Tivantinib With or Without Erlotinib Hydrochloride in Treating Patients With Metastatic or Locally Advanced Kidney Cancer That Cannot Be Removed by Surgery

Inclusion Criteria: * Patients must have histologically or cytologically confirmed papillary histology renal cell carcinoma which is metastatic, or locally advanced and unresectable; mixed histologies will be allowed provided that they contain \>= 50% of the papillary component * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension; x-rays, scans or physical examinations used for tumor measurement must have been completed within 28 days prior to registration; x-rays, scans or physical examinations for non-measurable disease must have been completed within 42 days prior to registration; all disease must be assessed and documented on the Baseline Tumor Assessment form * Patients with metastatic disease who have a resectable primary tumor and are deemed a surgical candidate may have undergone resection; at least 28 days must have elapsed since surgery and patient must have recovered from any adverse effects of surgery * Patients with a history of brain metastases who are asymptomatic and have not received steroid therapy in the 14 days prior to registration are eligible; anti-seizure medications are allowed provided they are non-enzyme inducing (e.g. topiramate, levetiracetam, gabapentin) * Patients may have received up to one prior systemic therapy for advanced or metastatic renal cell carcinoma; patients must not have received a MET inhibitor or erlotinib as prior therapy; at least 21 days must have elapsed since completion of prior systemic therapy, 42 days for nitrosoureas or mitomycin C; patients must have recovered from all associated toxicities at the time of registration * Patients may have received prior radiation therapy, but must have measurable disease outside the radiation port; at least 21 days must have elapsed since completion of prior radiation therapy; patients must have recovered from all associated toxicities at the time of registration * Patients must not be receiving or planning to receive any other investigational agents * Patients must have a complete physical examination and medical history within 28 days prior to registration * Patients must have a Zubrod performance status of 0-2 * White blood cell (WBC) \>= 2,000/mcL * Absolute neutrophil count (ANC) \>= 1,000/mcL * Platelet count \>= 75,000/mcL * Serum bilirubin =\< 1.5 x institutional upper limits of normal (ULN) * Serum glutamic oxaloacetic transaminase (SGOT)/aspartate aminotransferase (AST) and serum glutamic pyruvate transaminase (SGPT)/alanine aminotransferase (ALT) must be =\< 1.5 x the institutional ULN unless the liver is involved with the tumor, in which case serum transaminase (SGOT/SGPT) must be =\< 5 x the institutional ULN * Serum creatinine must be =\< 2 x the institutional ULN * Sodium, potassium and calcium must be obtained within 14 days prior to registration * Patients with a known history of the following corneal diseases are not eligible: dry eye syndrome, Sjogren's syndrome, keratoconjunctivitis sicca, exposure keratopathy, Fuchs' dystrophy or other active disorders of cornea * Patients known to be human immunodeficiency virus (HIV)-positive and receiving combination anti-retroviral therapy are not eligible * Patients must be able to take oral medications; patients must not have gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for intravenous (IV) alimentation, prior surgical procedures affecting absorption, or active peptic ulcer disease; patients with intractable nausea or vomiting are not eligible * Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures * No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease-free for 5 years * Patients must be offered the opportunity to participate in specimen banking for future translational medicine studies * All patients must be informed of the investigational nature of this study and must sign and give written informed consent * As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system