A Study of TAK-981 Given With Rituximab in Adults With Relapsed or Refractory CD20-Positive Non-Hodgkin Lymphoma

Inclusion Criteria: Each participant must meet all the following inclusion criteria to be enrolled in the study: 1. Participant Population: o. For Phase 1 Dose Escalation: o. aNHL including mantle cell lymphoma and DLBCL histologies such as transformed DLBCL from low-grade lymphoma (follicular or others), DLBCL associated with small-cell infiltration in bone marrow, B-cell lymphoma with intermediate features between DLBCL and Burkitt's lymphoma or with intermediate features between DLBCL and Hodgkin lymphoma, FL grade 3B, and aggressive B-cell lymphoma unclassifiable who must have previously received rituximab, cyclophosphamide, doxorubicin (hydroxydaunorubicin), vincristine, (Oncovin) and prednisone (R-CHOP) (or equivalent anti-CD20 containing therapy) and 1 additional line of therapy in the r/r setting. o. iNHL (including FL of grades 1-3A and marginal zone lymphoma) refractory to rituximab or to any other anti-CD20 monoclonal antibodies, who have received at least 1 prior systemic therapy for r/r iNHL. o. Rituximab or anti-CD20 refractoriness is defined as failure to respond to, or progression during, any previous rituximab/anti-CD20-containing regimen (monotherapy or combined with chemotherapy), or progression within 6 months of the last rituximab or anti-CD20 dose. Note: The minimum qualifying rituximab/anti-CD20 dose is 1 full cycle (that is, weekly\*4 doses monotherapy or 1 complete dose if combined with chemotherapy). Prior anti-CD20 antibody or cytotoxic drugs may have been administered as single agents or as components of combination therapies. Each repeated course of the same single-agent or combination is considered an independent regimen. o. For Phase 2, the following confirmed CD20+: o. r/r DLBCL progressed or relapsed after a prior CAR T-cells therapy that has received approval by a health authority for the treatment of DLBCL (Cohort A). o. r/r DLBCL that has progressed or relapsed after at least 2 but no more than 3 prior lines of systemic therapy and has not I prior cellular therapy. At least one prior line of therapy must have included a CD20-targeted therapy (Cohort B). o. r/r FL that has progressed or relapsed after at least 2 but no more than 3 prior lines of systemic therapy. At least 1 prior line of therapy must have included a CD20-targeted therapy (Cohort C). 2. Must be considered ineligible in the opinion of the investigator, or refused autologous stem-cell transplantation (ASCT). 3. Eastern Cooperative Oncology Group (ECOG) performance score of less than or equal to (\<=) 2. 4. Adequate bone marrow function per local laboratory reference range at screening as follows: o Platelet count greater than or equal to (\>=) 75.0\*10\^9/L, Grade 2 thrombocytopenia (platelet count \>=50.0\*10\^9 per liter \[/L\]) is allowed if it is clearly due to marrow involvement with no evidence of myelodysplastic syndrome or hypoplastic bone marrow if found. Absolute neutrophil count (ANC) \>=1.0\*10\^9/L. Hemoglobin \>=85 gram per liter (g/L) (red blood cell \[RBC\] transfusion allowed \>=14 days before assessment). 5. Adequate renal and hepatic function, per local laboratory reference range at screening as follows: * Calculated creatinine clearance \>=30 milliliter per minute (mL/min) calculated with Cockcroft-Gault formula. * Potassium levels \>=lower limit of normal (LLN). For potassium \>upper limit of normal (ULN) discussion with Takeda medical monitor (MM)/designee recommended. * Aspartate aminotransferase and alanine aminotransferase \<=3.0\*the ULN of the institution's normal range; bilirubin \<=1.5\*ULN. Participants with Gilbert's syndrome may have a bilirubin level \>1.5\*ULN, per discussion between the investigator and the medical monitor. 6. Left ventricular ejection fraction (LVEF) \>=40 percent (%); as measured by echocardiogram or multiple gated acquisition (MUGA) scan. 7. Suitable venous access for safe drug administration and the study-required PK and pharmacodynamic sampling. 8. Have at least 1 bidimensionally measurable lesion per Lugano Classification by computed tomography (CT). Tumor lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. 9. Willing to consent to 1 mandatory pretreatment and 1 on-treatment skin biopsy during Phase 1. The skin biopsy entry requirement may be discontinued by the sponsor once there is enough pharmacodynamic evidence of target engagement. 10. For participants enrolled in Phase 2, if available, mandatory submission of archival tumor tissue acquired ≤12 months prior to screening. 11. Recovered to Grade 1, baseline or established as sequela, from all toxic effects of previous therapy (except alopecia, neuropathy, autoimmune endocrinopathies with stable endocrine replacement therapy, neurotoxicity \[Grade 1 or 2 permitted\], or bone marrow parameters \[any of Grade 1, 2, permitted if directly related to bone marrow involvement\]). Exclusion Criteria: Participants meeting any of the following exclusion criteria are not to be enrolled in the study: 1. Central nervous system lymphoma; active brain or leptomeningeal metastases, as indicated by positive cytology from lumbar puncture or CT scan/magnetic resonance imaging (MRI). 2. History of Grade \>=3 infusion-related reaction (IRR) that lead to permanent discontinuation of previous rituximab treatment. 3. Post transplantation lymphoproliferative disease except relapsed NHL after ASCT. 4. Undergone ASCT or treatment with cellular therapy including CAR T within \<=12 weeks of TAK-981 dosing. 5. Prior allogeneic hematopoietic stem-cell transplantation. 6. Lymphomas with leukemic expression. 7. Prior anticancer therapy including chemotherapy, hormonal therapy, or investigational agents within 2 weeks or within at least 5 half-lives before TAK-981 dosing, whichever is shorter. Low dose steroids (oral prednisone or equivalent \<=20 mg per day), hormonal therapy for prostate cancer or breast cancer (in adjuvant situation), and treatment with bisphosphonates and receptor activator of nuclear factor kappa-B ligand (RANKL) inhibitors are allowed. 8. Major surgery within 14 days before the first dose of study drug and not recovered fully from any complications from surgery. 9. Significant medical diseases or conditions, as assessed by the Investigators and sponsor that would substantially increase the risk-benefit ratio of participating in the study. This includes but is not limited to acute myocardial infarction or unstable angina within the last 6 months; uncontrolled diabetes mellitus; significant active bacterial, viral, or fungal infections; severely immunocompromised state; severe non-compensated hypertension and congestive heart failure New York Heart Association Class III or IV; ongoing symptomatic cardiac arrhythmias of \>Grade 2, pulmonary embolism, or symptomatic cerebrovascular events; or any other serious cardiac condition (example, pericardial effusion or restrictive cardiomyopathy). Chronic atrial fibrillation on stable anticoagulant therapy is allowed. 10. Known chronic hepatitis C and/or positive serology (unless due to vaccination or passive immunization due to immunoglobulin \[Ig\] therapy) for chronic hepatitis B. Known Human Immunodeficiency Virus (HIV) infection. 11. Second malignancy within the previous 3 years, except treated basal cell or localized squamous skin carcinomas, localized prostate cancer, cervical carcinoma in situ, resected colorectal adenomatous polyps, breast cancer in situ, or other malignancy for which the participant is not on active anticancer therapy. 12. Receipt of any live vaccine within 4 weeks of initiation of study treatment. 13. Active, uncontrolled autoimmune disease requiring \>20 mg of prednisone or equivalent, cytotoxics or biologicals. 14. Corticosteroid use within 1 week before the first dose of study drug, except as indicated for other medical conditions such as inhaled steroid for asthma, topical steroid use, or as premedication for administration of study drug or contrast. Participants requiring steroids at daily doses \>20 mg prednisone equivalent systemic exposure daily, or those who are administered steroids for lymphoma control or white blood cell count lowering are not eligible. 15. With baseline prolongation of the QT interval with Fridericia correction method (QTcF) (example, \>470 milliseconds (ms) for women and \>450 ms for men and a history of congenital long QT syndrome, or torsades de pointes). 16. Receiving or requiring the continued use of medications that are known to be strong or moderate inhibitors and inducers of Cytochrome P450 3A4/5 (CYP3A4/5) and strong P-glycoprotein (Pgp) inhibitors. To participate in this study, such participants should discontinue use of such agents for at least 2 weeks (1 week for CYP3A4/5 and Pgp inhibitors) before receiving a dose of TAK-981. 17. Participants in Germany who are committed to an institution by virtue of an order issued either by judicial or administrative authorities as per German law.