Daratumumab for Relapsed/Refractory Primary Effusion Lymphoma, Plasmablastic Lymphoma, and Multicentric Castleman Disease

* INCLUSION CRITERIA: * Participants with histologically confirmed primary effusion lymphoma (PEL) including extracavitary and KSHV-associated large cell lymphoma variants, plasmablastic lymphoma (PBL), and/or KSHV-associated multicentric Castleman disease (MCD) that has relapsed, and/or is refractory after frontline chemotherapy, or who are ineligible for front-line chemotherapy * Age \>= 18 years. * Any HIV status * Participants with HIV must be receiving or will initiate an effective combination antiretroviral therapy (ART) regimen and must have an undetectable HIV VL which is defined as \<200 copies/mL. * Participants with PEL or PBL must meet the following criteria: * Must have measurable or assessable lymphoma * ECOG performance status (PS) 0-2 or 3 if secondary to PEL or PBL * Adequate hematological and renal functions as defined below: * Hemoglobin (Hgb) \> 7 g/dL * Creatinine clearance (CrCl) \>= 15 mL/min/1.73 m\^2 * Must have received first-line curative-intent therapy (anthracycline-containing chemotherapy) for PEL or PBL, unless such therapy is contraindicated due to infection that precludes combination chemotherapy (such as progressive multifocal leukoencephalopathy) or if there is a contraindication to receiving CHOP or EPOCH (such as multi-organ failure). * Participants with KSHV-MCD must meet the following criteria: * ECOG performance status (PS) 0-2 or 3 if secondary to MCD * Adequate hematological and renal functions as defined below: * Hemoglobin (Hgb) \> 7 g/dL * Creatinine clearance (CrCl) \>= 15 mL/min/1.73 m\^2 * At least one clinical symptom attributed to KSHV-MCD * Fever (\>38 degrees Celsius) * Fatigue * Gastrointestinal symptoms * Respiratory/sinus symptoms * Rash * At least one laboratory abnormality attributed to KSHV-MCD * Anemia (Hgb \[men\] \< 12.5 g/dL, Hgb \[women\] \< 11 g/dL) * Thrombocytopenia (\< 150 K/microL) * Hypoalbuminemia (\< 3.4 g/dL) * Hyponatremia (\< 135 mmol/L) * Elevated C-reactive protein (CRP) (\> 3 mg/L) * For participants with evidence of chronic hepatitis B virus (HBV) infection, participants must be on suppressive therapy with an undetectable VL. * Participants who are seropositive for hepatitis C virus (HCV)are eligible only in the setting of a sustained virologic response \[SVR\], defined as aviremia, at least 12 weeks after completion of antiviral therapy. * Participants that have received investigational agents on other clinical trials must have had a washout period of 2 weeks or 5 drug half-lives, whichever is longer. * Women of child-bearing potential (WOCBP) must agree to use an effective (dual) form of contraception (barrier, surgical sterilization, abstinence) prior to study entry and for the duration of study participation and for 3 months after the last dose of study drug. WOCBP must refrain from egg donations during the study and for 3 months after the last dose of daratumumab. * Men must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and up to 3 months after the last dose of the study drug(s). We also will recommend men with female partners of childbearing potential to ask female partners to be on an effective birth control (hormonal, intrauterine device \[IUD\], surgical sterilization). * Breastfeeding participants must be willing to discontinue breastfeeding from study treatment initiation through 3 months after the last dose of the study drug. * Participants must understand and sign a written informed consent document. EXCLUSION CRITERIA: * Participants who have had anticancer treatment within the last 2 weeks unless the cancer treatment is for a malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial, such as local treatment for carcinoma in situ or hormonal therapy for prostate or breast carcinoma. Toxicity related to prior therapies other than hair loss and neuropathy must have resolved to grade 1. * KS requiring urgent treatment with cytotoxic chemotherapy. * Bilirubin (total) \> 1.5 times the upper limit of normal; aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) \> 3 times the upper limit of normal (ULN); EXCEPTIONS: * Total bilirubin \>= 5 mg/dL in participants with Gilbert's syndrome as defined by \> 80% unconjugated * If the elevated total bilirubin or AST/ALT are due to ART or lymphoma * ANC \< 1000/mm\^3 and platelets \< 75,000/mm\^3 unless related to lymphoma and/or KSHV-MCD or prior therapy. * No life-threatening or organ-threatening manifestations of KSHV-MCD. * Clinically significant cardiac disease, including: * Myocardial infarction within 6 months of randomization, or an unstable or uncontrolled disease/condition related to or affecting cardiac function (e.g., unstable angina, congestive heart failure, New York Heart Association Class IIIIV). * Uncontrolled cardiac arrhythmia * Chronic obstructive pulmonary disease (COPD) with a forced expiratory volume in 1 second (FEV1) \< 50% of predicted normal. * Moderate or severe persistent asthma within the past 2 years, or uncontrolled asthma of any classification. Note that participants who currently have controlled intermittent asthma or controlled mild persistent asthma are allowed to participate in the study. * Pregnant people as evaluated by a positive serum or urine beta-human chorionic gonadotropin (Beta-hCG) test * Participants with severe uncontrolled intercurrent illness, evaluated by history, physical exam and chemistry panel. Participants with severe intercurrent illnesses attributed to lymphoma may be eligible per PI s or designee s discretion.