A Study of TAK-186 (Also Known as MVC-101) in Adults With Advanced or Metastatic Cancer

Key Inclusion Criteria: * Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 1 * Ability to provide informed consent and documentation of informed consent before initiation of any study-related tests or procedures that are not part of standard of care for the participant's disease. Participants must also be willing and able to comply with study procedures, including the acquisition of specified research specimens. * Life expectancy ≥ 12 weeks * Measurable disease as per Response Evaluation Criteria in Solid Tumors (RECIST) v1.1 criteria and documented by Computed tomography (CT) and/or magnetic resonance imaging (MRI). The definitions for measurable lesions are the same whether conventional and modified RECIST criteria are applied. Cutaneous or subcutaneous lesions must be measurable by calipers. Lesions to be used as measurable disease for the purpose of response assessment must either a) not reside in a field that has been subjected to prior radiotherapy, or b) have demonstrated clear evidence of radiographic progression since the completion of prior radiotherapy and before study enrollment or c) have been radiated at least 6 months before study enrollment. * Tumor Histology Types: * Participants with pathologically proven, unresectable, locally advanced or metastatic solid tumors that based on literature reports are considered to express EGFR. During cohort expansion, participants with locally advanced or metastatic solid tumors expressing EGFR including advanced or metastatic NSCLC, CRC, and HNSCC are eligible for enrollment. \* Tumors During Cohort Expansion: * Participants with pathologically proven, unresectable, locally advanced or metastatic solid tumors that based on literature reports are considered to express EGFR are eligible for enrollment: * NSCLC: locally advanced or metastatic NSCLC that has progressed during or following treatment with platinum-based chemotherapy, a checkpoint inhibitor (unless known to be PD-L1 negative), or targeted therapy (for participants with a known actionable mutation). * CRC: locally advanced or metastatic CRC that has progressed after systemic therapies, including irinotecan, oxaliplatin, an anti-EGFR inhibitor (if K-RAS or N-RAS is WT), a checkpoint inhibitor (if MSI-H), and a VEGF inhibitor (if locally approved and accessible as a standard-of-care). * HNSCC: HNSCC that has progressed during or following treatment with a checkpoint inhibitor (unless ineligible, e.g, PD-L1 negative) and platinum-based chemotherapy (unless ineligible for or intolerant to platinum-based chemotherapy) with or without cetuximab for metastatic or recurrent disease. 1. Participants with salivary gland tumors will not be considered as having HNSCC. 2. Participants who refuse surgery for potentially curable disease where the surgery or radiotherapy could result in severe morbidity are eligible. The reason for the refusal will be captured in the electronic case report form (eCRFs). * Archival Tissue: * Participants must allow acquisition of existing formalin-fixed paraffin-embedded (FFPE) archival tumor sample, either a block or unstained slides. Participants who provide fresh pretreatment biopsy samples will not be required to submit archival tumor samples. * Tumor Biopsy: • Participants must be willing to consent to mandatory pretreatment (during screening) and on-treatment fresh tumor biopsies for cohort expansion phase and backfill in dose escalation. Once the target number of biopsies have been collected, additional paired pretreatment and on-treatment biopsies will not be required; sample collection will be optional after this time point. For fresh tumor biopsies, the lesion must be accessible (those occurring outside the brain or those that are accessible by an interventional or endoscopic procedure) for a low-risk biopsy procedure that does not place the participant at an unjustifiable risk in the opinion of the investigator. Participants who have an archived biopsy specimen available that was obtained up to 90 days prior to treatment initiation and have received no other treatment from the time of biopsy until the start of treatment with TAK-186, may submit that archived specimen in lieu of a pretreatment biopsy upon agreement from the sponsor. * Laboratory Features: * Acceptable laboratory parameters as follows: 1. Albumin ≥ 3.0 g/dL 2. Platelet count ≥ 75 × 103/μL 3. Hemoglobin ≥ 9.0 g/dL 4. Absolute neutrophil count ≥ 1.0 × 103/μL 5. Alanine aminotransferase (ALT)/aspartate aminotransferase (AST) ≤ 3.0 × upper limit of normal (ULN); for participants with hepatic metastases, ALT/AST ≤ 5 × ULN 6. Total bilirubin ≤ 1.5 × ULN, except participants with Gilbert's syndrome, who may enroll if the conjugated bilirubin is within normal limits. 7. Creatinine clearance of ≥ 30 mL/minute using Cockcroft-Gault equation. * Reproductive Features: * WOCBP must have a negative serum pregnancy test performed within 72 hours before the initiation of study drug administration. WOCBP must use 1 form of highly effective method and 1 additional effective (barrier) method of contraception at the same time throughout the study, starting at screening through 90 days after the last dose of TAK-186. Contraception methods may be considered highly effective if they can achieve a failure rate of less than 1% per year when used consistently and correctly. * Male participants with partners of childbearing potential must use barrier contraception during the entire study treatment period through 120 days after the last dose of study drug and must not donate sperm during this period. In addition, male participants should also have their partners use contraception (as documented for female participants) for the same period of time. \* Previous Checkpoint Inhibitor Therapy: * Participants who have previously received an immune checkpoint before enrollment must have checkpoint inhibitor immune-related toxicity resolved to either Grade ≤ 1 or baseline * Symptomatic central nervous system (CNS) metastases must have been treated, be asymptomatic for ≥ 14 days, and meet the following criteria at the time of enrollment: 1. No concurrent treatment for CNS disease (e.g., surgery, radiation, corticosteroids ≥ 10 mg prednisone per day or equivalent). 2. No concurrent leptomeningeal disease or spinal cord compression. Key Exclusion Criteria: * Participants with a history of known autoimmune disease with the exceptions of: 1. Vitiligo. 2. Psoriasis not requiring systemic treatment for \> 1 year before receiving TAK-186. 3. History of Graves' disease in participants now euthyroid for \> 4 weeks. 4. Hypothyroidism managed by thyroid replacement. 5. Alopecia. 6. Well-controlled diabetes type 1. * Major surgery or traumatic injury within 8 weeks before first dose of TAK-186. * Unhealed wounds from surgery or injury. * Radiation therapy \< 2 weeks before initiation of TAK-186. * Treatment with \> 10 mg per day of prednisone (or equivalent) or other immune-suppressive drugs within the 7 days before the initiation of study drug. Steroids for topical, ophthalmic, inhaled, or nasal administration are allowed. * Prior therapy within the following timeframe before the planned start of TAK-186 as follows: 1. Cytotoxic chemotherapy, small molecule inhibitors, radiation, interventional radiology procedure, or similar investigational therapies: ≤ 2 weeks or 5 half-lives, whichever is shorter. 2. Monoclonal antibodies, antibody-drug conjugates, radioimmunoconjugates, or similar investigational therapies: ≤ 4 weeks. 3. Concurrent use of hormones either to maintain castrate levels of testosterone in participants with castration-sensitive prostate cancer or for non-cancer-related conditions (e.g., insulin for diabetes, hormone replacement therapy) is acceptable. Bisphosphonates are permitted for supportive care of bone metastases (e.g., breast or prostate cancer) or osteoporosis. * Clinically significant cardiovascular or vascular disease including: 1. Myocardial infarction or unstable angina \< 6 months before the initiation of study drug. 2. Clinically significant cardiac arrhythmia (e.g., with potential for hemodynamic instability). 3. Uncontrolled hypertension: systolic blood pressure \> 180 mmHg; diastolic blood pressure \> 100 mmHg. 4. Pulmonary embolism, stroke, or transient ischemic attack \< 6 months before initiation of TAK-186. 5. QTcF (QT interval by Fridericia correction) prolongation \> 480 msec. 6. Congestive heart failure (New York Heart Association Class III or IV). 7. Pericarditis or clinically significant pericardial effusion. 8. Myocarditis. 9. Vasculitis not resolved \< 6 months before TAK-186 initiation. * Clinically significant gastrointestinal disorders including: 1. Gastrointestinal perforation \< 6 months before study drug administration. Participants must have documented evidence (e.g., upper endoscopy, colonoscopy) of completely healed area of prior perforation. 2. Gastrointestinal bleeding \< 2 months before study drug administration. Participants must have documented evidence (e.g., upper endoscopy, colonoscopy) of completely healed area of prior bleeding. 3. Pancreatitis \< 6 months before the initiation of study drug. Participants must have a CT scan negative for evidence of remaining disease or normal pancreatic enzyme levels for \> 4 weeks before the initiation of TAK-186. 4. Diverticulitis flare \< 2 months before study drug administration. Participants must have a CT scan negative for evidence of remaining disease before the initiation of TAK-186. 5. History of Crohn's disease or ulcerative colitis. * Inflammatory process that has not resolved for ≥ 4 weeks from the date of first study dose. Participants with chronic low-grade inflammatory processes such as radiation-induced pneumonitis are excluded regardless of duration. * Clinically significant pulmonary compromise (e.g., requirement for supplemental oxygen on a continuous basis). * Active viral, bacterial, or systemic fungal infection requiring parenteral treatment within 7 days before the initiation of study drug. Systemic antiviral, antifungal, or antibacterial therapy must be completed \> 1 week before the initiation of study drug. Antimicrobial prophylaxis (e.g., for Pneumocystis carinii infection) may continue the antimicrobial for that purpose. * Vaccination with any live virus vaccine within 4 weeks before the initiation of study drug administration or vaccination with other vaccines 2 weeks before the initiation of study drug administration. Inactivated annual influenza vaccination is allowed. * Participants who are known to be human immunodeficiency virus positive or who are known to be hepatitis B or C positive. Participants treated for hepatitis C must have viral titers of 0 for ≥ 2 years to be eligible. Participants with hepatitis B having undetectable or ≤ 500 IU hepatitis B viral titers are eligible. Participants with hepatocellular carcinoma (HCC) known history of hepatitis B are excluded, regardless of hepatitis B viral titers. * Second primary invasive malignancy not in remission for ≥ 3 years. Exceptions include non-melanoma skin cancer, cervical carcinoma in situ, localized prostate cancer (Gleason score ≤ 7), resected melanoma in situ, or any malignancy considered to be indolent and never having required therapy, excluding indolent lymphoid malignancies. * Any serious underlying medical or psychiatric condition that would preclude understanding and rendering of informed consent or impair the ability of the participant to receive or tolerate the planned treatment. * Known hypersensitivity to TAK-186 (or any excipient \[trehalose, histidine, arginine, or polysorbate-80\] contained in the drug or diluent formulation) known hypersensitivity to tocilizumab. * Investigative site personnel or sponsor personnel directly affiliated with this study or known hypersensitivity to tocilizumab. * Prisoners or other individuals who are involuntarily detained. * Any medical or non-medical issue that would contraindicate the participant's participation in the study or confound the results of the study. * Female participants who are breastfeeding.