LN-145 or LN-145-S1 in Treating Patients With Relapsed or Refractory Ovarian Cancer, Triple Negative Breast Cancer (TNBC), Anaplastic Thyroid Cancer, Osteosarcoma, or Other Bone and Soft Tissue Sarcomas

Inclusion Criteria: * Age between 18 and 70 (Subjects aged 16-70 may be enrolled into the osteosarcoma cohort). * Subjects must be willing and able to provide informed consent. For patients \< 18 years of age, their parents or legal guardians must sign a written informed consent. Assent, when appropriate, will be obtained according to institutional guidelines. * Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0 or 1. * Subjects must have an area of tumor amenable to excisional biopsy (core biopsies may be allowed as detailed in protocol) for the generation of TIL separate from, and in addition to, a target lesion to be used for response assessment. * Any prior therapy directed at the malignant tumor, including radiation therapy, chemotherapy, and biologic/targeted agents must be discontinued at least 28 days prior to enrollment for preparing TIL therapy. Palliative therapy may be received during the screening period with principal investigator (PI) approval for lesions that are not expected to be used for TIL generation or as target lesions. * Absolute neutrophil count (ANC) \>= 1000/mm\^3 (within 7 days of enrollment). * Hemoglobin \>= 8.0 g/dL (transfusion allowed) (within 7 days of enrollment). * Platelet count \>= 100,000/mm\^3 (within 7 days of enrollment). * Alanine aminotransferase (ALT)/serum glutamate pyruvate transaminase (SGPT) and aspartate aminotransferase (AST)/serum glutamic-oxaloacetic transaminase (SGOT) =\< 2.5 x the upper limit of normal (ULN) * Patients with liver metastases may have liver function tests (LFT) =\< 5.0 x ULN (within 7 days of enrollment). * Calculated creatinine clearance (Cockcroft-Gault) \>= 50.0 mL/min (within 7 days of enrollment). * Total bilirubin =\< 1.5 x ULN (within 7 days of enrollment). * Prothrombin time (PT) \& activated partial thromboplastin time (aPTT) =\< 1.5 x ULN (correction with vitamin K allowed) unless subject is receiving anticoagulant therapy (which should be managed according to institutional norms prior to and after excisional biopsy) (within 7 days of enrollment) * Negative serum pregnancy test (female subjects of childbearing potential) (within 7 days of enrollment) * Subjects must not have a confirmed human immunodeficiency virus (HIV) infection. * Subjects must have a 12-lead electrocardiogram (EKG) showing no active ischemia and Fridericia's corrected QT interval (QTcF) less than 480 ms. * Subjects must also have a negative dobutamine stress echocardiogram. If neither of these tests can be performed, subject may complete alternative cardiac evaluation deemed clinically appropriate based on the recommendation of cardiology. * Subjects of childbearing potential must be willing to practice an approved highly effective method of birth control starting at the time of informed consent and for 1 year after the completion of the lymphodepletion regimen. Approved methods of birth control are as follows: * Hormonal contraception (i.e. birth control pills, injection, implant, transdermal patch, vaginal ring). * Intrauterine device (IUD). * Tubal ligation or hysterectomy. * Subject/partner status post vasectomy. * Implantable or injectable contraceptives. * Condoms plus spermicide. * Able to adhere to the study visit schedule and other protocol requirements. * Pulmonary function tests (spirometry) demonstrating forced expiratory value (FEV) 1 greater than 65% predicted or forced vital capacity (FVC) greater than 65% of predicted. * Ovarian cancer cohort only: Subjects must have high-grade non-mucinous histology (carcinosarcomas are allowed). * Ovarian cancer cohort only: Subjects must have failed at least two prior lines of chemotherapy (i.e. frontline adjuvant chemotherapy plus one additional line for recurrent/progressive disease), and have platinum resistant disease. * TIL-ICI ovarian cancer cohort only: Same as "ovarian cancer" above. * TIL-ICI ovarian cancer cohort only: Enrolled after activation of protocol version 2.0. * Osteosarcoma cohort only: Subjects with osteosarcomas must have relapsed or become refractory to conventional therapy and have received a regimen including some combination of high-dose methotrexate, doxorubicin, cisplatin, and/or ifosfamide. * Other bone and soft tissue sarcomas cohort only: Subjects with dedifferentiated chondrosarcomas, dedifferentiated giant cell tumor of bone, giant cell tumor of bone, undifferentiated pleomorphic sarcoma of bone, or high-grade unclassified sarcomas of bone must have received at least one prior line of therapy unless no standard first-line therapy exists in which case enrollment as initial therapy is allowed. * Other bone and soft tissue sarcomas cohort only: Subjects with other soft tissue sarcomas who have received at least one line of therapy. * TIL-ICI sarcoma cohort only: Subjects with osteosarcomas must have relapsed or become refractory to conventional therapy and have received a regimen including some combination of high-dose methotrexate, doxorubicin, cisplatin, and/or ifosfamide. * TIL-ICI sarcoma cohort only: Subjects with dedifferentiated chondrosarcomas, dedifferentiated giant cell tumor of bone, giant cell tumor of bone, undifferentiated pleomorphic sarcoma of bone, or high-grade unclassified sarcomas of bone must have received at least one prior line of therapy unless no standard first-line therapy exists in which case enrollment as initial therapy is allowed. * TIL-ICI sarcoma cohort only: Subjects with other soft tissue sarcomas who have received at least one line of therapy. * TIL-ICI sarcoma cohort only: Enrolled after activation of protocol version 2.0. * Anaplastic and poorly-differentiated thyroid cancer cohort only: Pathologic findings supporting the clinical impression of anaplastic thyroid carcinoma. Diagnosis may include consistent with, or suggestive of terminology associated with: anaplastic thyroid carcinoma, undifferentiated carcinoma, squamous carcinoma; carcinoma with spindled, giant cell, or epithelial features; poorly differentiated carcinoma. * Anaplastic and poorly-differentiated thyroid cancer cohort only: Measurable distant metastatic disease by RECIST v1.1. * Anaplastic and poorly-differentiated thyroid cancer cohort only: Subjects who are planned for surgical resection of their tumor, or subjects who are planned for surgery to stabilize the airway (i.e., tracheostomy). Subjects who have a stable airway at the time of consent are eligible if there is tumor that can be resected for TIL manufacturing. * Anaplastic and poorly-differentiated thyroid cancer cohort only: Previous external beam radiation to the neck is allowed as long as there is a measurable lesion that can be biopsied (separate from the area of radiated tumor) and at least one other for RECIST response assessment. * TIL-ICI-TNBC cohort only: Subjects must have a confirmed diagnosis of metastatic triple negative breast cancer (Stage IV or recurrent) histologically confirmed as per the AJCC staging system. * Subjects must have TNBC, defined from standard pathologic assays as negative for ER and PR (\<10% tumor staining) and negative for HER2 (IHC\<3, gene copy number not amplified; per ASCO/CAP guidelines). * Subjects must have had at least 1 and no more than 3 prior lines of systemic anticancer therapy for metastatic disease. * The resectable lesion must be a minimum of 1.5 cm in diameter. Exclusion Criteria: * Active or uncontrolled intercurrent illness, including but not limited to ongoing or active infections, coagulation disorders or other major medical illnesses of the cardiovascular, respiratory or immune system. Principal investigator (PI) or his/her designee shall make the final determination regarding appropriateness of enrollment. * Patients with active viral hepatitis. * Patients who have a left ventricular ejection fraction (LVEF) \< 45% at screening. * Patients with a history of prior adoptive cell therapies. * Persistent prior therapy-related toxicities greater than grade 2 according to Common Toxicity Criteria for Adverse Events (CTCAE) version (v)4.03, except for peripheral neuropathy, alopecia, or vitiligo prior to enrollment. * Patients who have had a documented grade 2 or greater diarrhea or colitis due to previous immunotherapy must have been asymptomatic for at least 6 months or had a normal colonoscopy post checkpoint treatment, by visual assessment, prior to enrollment. * Patients with immunotherapy-related endocrinopathies (e.g. hypothyroidism or hypopituitarism, who are stable on hormonal substitution) and controlled with hormonal replacement, are allowed. * Primary immunodeficiency. * History of organ or hematopoietic stem cell transplant. * Chronic steroid therapy, however prednisone or its equivalent is allowed at =\< 10 mg/day. * Patients who are pregnant or nursing. * Presence of a significant psychiatric disease, which in the opinion of the principal investigator or his/her designee, would prevent adequate informed consent. * History of clinically significant autoimmune disease including active, known, or suspected autoimmune disease. Subjects with resolved side effects from prior checkpoint inhibitor therapy, vitiligo, psoriasis, type 1 diabetes or resolved childhood asthma/atopy would be an exception to this rule. Subjects that require intermittent use of bronchodilators or local steroid injections would not be excluded. Subjects with hypothyroidism stable on hormone replacement or Sjogren's syndrome will not be excluded. * History of clinically significant chronic obstructive pulmonary disease (COPD), asthma, interstitial lung disease, or other chronic lung disease. * History of a second malignancy (diagnosed in the last 5 years). Exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy. For the thyroid cancer cohort, subjects have a higher likelihood of prior cancers. Therefore, for this cohort, history of prior thyroid cancer or other indolent cancers is not considered exclusionary if in the opinion of the treating physician such cancers are indolent or unlikely to affect the overall prognosis based on the active thyroid cancer. * History of known active central nervous system metastases and/or carcinomatous meningitis. Subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to initiation of lymphodepletion. * Has received a live vaccine within 30 days prior to the initiation of lymphodepletion. * Patients who have a contraindication to or history of hypersensitivity reaction to any components or excipients of the TIL therapy or the other study drugs: non-myeloablative-lymphodepletion (NMA-LD) (cyclophosphamide, mesna, and fludarabine); IL-2; antibiotics of the aminoglycoside group (i.e., streptomycin, gentamicin); any component of the TIL infusion product formulation including human serum albumin (HSA), IL-2, and dextran-40, nivolumab or ipilimumab. * Any other condition that in the investigator's judgement would significantly increase the risks of participation.