Dose Escalation and Efficacy Study of mRNA-2416 for Intratumoral Injection Alone and in Combination With Durvalumab for Participants With Advanced Malignancies

Inclusion Criteria: * Written informed consent prior to completing any study-specific procedure * Dose Escalation and Dose Confirmation Periods: Histologically- or cytologically-confirmed advanced/metastatic solid tumor or lymphoma by pathology report and who has received, or been intolerant to, all approved therapies * Dose Expansion Period: Histologically or cytologically confirmed diagnosis of: epithelial cancer of the ovary, fallopian tube, or peritoneum which is platinum resistant or platinum refractory. Participants must have received at least 2 prior lines of therapy. Participants with known Breast Cancer gene 1 (BRCA) mutation positive must have been treated with and progressed on at least 1 prior poly\[ADP-ribose\] polymerase inhibitor (PARPi) * Lesions for intratumoral injection and biopsies: * Dose Escalation: A minimum of one lesion that is easily accessible for injection where easily accessible is defined as a cutaneous or subcutaneous mass that is palpable and/or visualizable by ultrasound * Dose Confirmation: A minimum of one visceral lesion injectable with ultrasound or computer tomography (CT) guidance and that is not encasing or abutting major vascular structures or are in a location that are considered high risk for AEs by the enrolling physician * Dose Expansion: A minimum of one lesion amenable to injection (either non-visceral or visceral). Participants must have a tumor lesion amenable to biopsy and consent to a pre-treatment and an on-treatment biopsy. For participants with only one lesion amenable to injection, biopsy, and RECIST assessment, the lesion must be ≥2 centimeters (cm) * Biopsy Cohort Enrichment: Participants must have a tumor lesion amenable to biopsy and consent to a pre-treatment and an on-treatment biopsy * All lesion(s) targeted for the initial injection must be ≥0.5 cm on longest diameter, be at least 5 mm thick, and have distinct borders based on exam or imaging, not close to critical structures such as major vessels, nerves, or airways * Participants must have measurable disease as determined by RECIST v1.1 (solid tumors) or Cheson 2014 criteria (lymphomas). \- Dose Expansion: Participants must have at least 1 measurable lesion per RECIST v1.1 which has not been previously irradiated * Eastern Cooperative Oncology Group (ECOG) performance status of ≤1 * Adequate hematological and biological function * Adequate thyroid function: Thyroid-stimulating hormone within normal range. * Female participants of childbearing potential must have a negative serum pregnancy test during screening. * Male and female participants must agree to use a highly reliable method of birth control. * Must have life expectancy of at least 12 weeks * Body weight \>30 kilograms (kg) Exclusion Criteria: * Active central nervous system tumors or metastases * Treatment with chemotherapy, radiation (local radiation for palliative care is permitted), hormonal anti-cancer treatment, or biologic therapy \<14 days prior to the first day of study treatment (Cycle 1 Day 1 \[C1D1\]). Treatment with any other investigational agent or treatment with any anti-cancer monoclonal antibody, immunostimulant, or vaccine \<28 days prior to C1D1 * Any unresolved toxicity NCI CTCAE Grade ≥2 from previous anticancer therapy with the exception of alopecia, vitiligo, and the laboratory values defined in the inclusion criteria * Participants with Grade ≥2 neuropathy will be evaluated on a case-by-case basis after consultation with the Study Physician * Participants with irreversible toxicity not reasonably expected to be exacerbated by the treatment with durvalumab may be included only after consultation with the Study Physician * Has active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease \[for example, colitis or Crohn's disease\], diverticulitis \[with the exception of diverticulosis\], systemic lupus erythematosus, Sarcoidosis syndrome, or Wegener syndrome \[for example, granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis, hypophysitis, uveitis\]). The following are exceptions to this criterion: * Participants with vitiligo or alopecia * Participants with hypothyroidism (for example, following Hashimoto syndrome) stable on hormone replacement * Participants with any chronic skin condition that does not require systemic therapy * Participants without active disease in the last 5 years may be included but only after consultation with the Moderna medical monitor * Participants with celiac disease controlled by diet alone * Has a history of primary immunodeficiency, allogenic solid organ transplantation, or tuberculosis * Receipt of live attenuated vaccine within 30 days prior to the first dose of study treatment. Note: Participants, if enrolled, should not receive live vaccine whilst receiving study treatment and up to 30 days after the last dose of study treatment. * History of human immunodeficiency virus infection * Active/chronic hepatitis B or C * Any of the following cardiac abnormalities: * Medically uncontrolled hypertension * New York Heart Association Class III or IV cardiac disease * Myocardial infarction within prior 6 months * Unstable angina * Unstable arrhythmias or mean QT interval corrected for heart rate using Fridericia's formula (QTcF) ≥470 milliseconds (ms) calculated from 3 electrocardiograms (ECGs) (within 15 minutes at 5 minutes apart) * History of another primary malignancy except for: * Malignancy treated with curative intent and with no known active disease ≥5 years before the first dose of IP and of low potential risk for recurrence * Adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease * Adequately treated carcinoma in situ without evidence of disease * Females who are pregnant or breastfeeding * Any other unstable or clinically significant concurrent medical condition (for example, substance abuse, psychiatric illness/social situations, uncontrolled intercurrent illness including active infection, arterial thrombosis, symptomatic pulmonary embolism, active interstitial lung disease, serious chronic gastrointestinal conditions associated with diarrhea) that would, in the opinion of the investigator, jeopardize the safety of a participant, impact their expected survival through the end of the study participation, and/or impact their ability to give written informed consent or comply with the protocol * For participants who have received prior anti-programmed death 1 (PD-1) or anti-programmed death ligand 1 (PD-L1) therapy, a participant must not have experienced any of the following: * Must not have experienced a toxicity that led to permanent discontinuation of prior immunotherapy. * All AEs while receiving prior immunotherapy must have completely resolved or resolved to baseline prior to screening for this study. * Must not have experienced a Grade ≥3 immune related AE or an immune related neurologic or ocular AE of any grade while receiving prior immunotherapy. Note: Participants with endocrine AEs of Grade ≤2 are permitted to enroll if they are stable while maintained on appropriate replacement therapy and are asymptomatic. * Must not have required the use of additional immunosuppression other than corticosteroids for the management of an AE, not have experienced recurrence of an AE if re-challenged, and not currently require maintenance doses of \>10 milligrams (mg) prednisone or equivalent per day. * Has an active infection including tuberculosis (clinical evaluation that includes clinical history, physical examination and radiographic findings, and tuberculosis testing in line with local practice), hepatitis B (known positive HBV surface antigen \[HBsAg\] result), hepatitis C, or human immunodeficiency virus (positive HIV 1/2 antibodies). Participants with a past or resolved HBV infection (defined as the presence of hepatitis B core antibody \[anti-HBc\] and absence of HBsAg) are eligible. Participants positive for hepatitis C (HCV) antibody are eligible only if polymerase chain reaction is negative for HCV RNA. * Has a history of (non-infectious) pneumonitis that required steroids or has current pneumonitis. * Has a history of leptomeningeal carcinomatosis. * Has involvement in the planning and/or conduct of the study. * Must not plan to donate blood or blood components while participating in this study and through 90 days after the last dose of study treatment