Akt Inhibitor MK2206 in Combination With Lapatinib Ditosylate in Patients With Advanced or Metastatic Solid Tumors or Breast Cancer

Inclusion Criteria: * Patients must have a histologically or cytologically confirmed advanced or metastatic solid tumor for which no standard curative measure exists * Patients must have either evaluable or measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan * Patients may have previously had disease progression on lapatinib, but should not have demonstrated prior serious or life-threatening intolerance to doses of lapatinib exceeding 1000 mg per day * Life expectancy of greater than 12 weeks * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (or Karnofsky \>= 60%) * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< upper limit of normal (ULN); in the case of a patient with known Gilbert's disease, s/he will be eligible as long as total serum bilirubin is less than 1.5 x ULN * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 times upper limit of normal * Creatinine =\< ULN OR creatinine clearance \>= 60 mL/min/1.73 m\^2 by Cockcroft-Gault for patients with creatinine levels above institutional normal * Patients with treated, stable brain metastases are allowed to enroll; patients must be at least 4 weeks from radiation and off any medications used to treat brain metastases; patients are allowed to be on anti-epileptic medications that are not metabolized by cytochrome P450; patients with brain metastases must have stable brain imaging within 4 weeks prior to starting study * Patients with progressive brain metastases who are not candidates for further local therapy (e.g. more radiation or surgery) but who have clinically asymptomatic brain are also eligible to enroll, as long as predicted life expectancy with the brain metastases meets or exceeds study requirements * Women of childbearing potential and men should use contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation AS WELL AS for one month after stopping use of the study agents * Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, the patient should inform the treating physician immediately * Ability to understand and the willingness to sign a written informed consent document * PART 2A: Patients must have histologically or cytologically documented locally advanced and unresectable OR metastatic breast cancer * PART 2A: Patients must have HER2+ cancer as defined by either: (a) 3+ for HER2 by immunohistochemistry (IHC), or (b) fluorescence in situ hybridization (FISH) or in situ hybridization (ISH) mean locus-to-centromeric ratio greater than or equal to 2.2; these analyses must be determined on an invasive component of the cancer at either the primary site or the metastatic site * PART 2A: Patients must have previously received trastuzumab, either in the adjuvant or metastatic setting * PART 2A: All patients in this cohort must have archived primary or metastatic tissue blocks available Exclusion Criteria: * Patients who have had chemotherapy, therapy with trastuzumab, bevacizumab or other targeted therapy, or radiotherapy within 4 weeks (6 weeks for regimens including carmustine \[BCNU\], nitrosoureas or mitomycin C) prior to entering the study; the following will apply with regards to endocrine therapy: * Patients receiving an aromatase inhibitor (AI) are eligible as long as they stop the AI one day prior to beginning study agents * Patients receiving tamoxifen or fulvestrant should have received their last dose at least 2 weeks prior to beginning study agents * Patients who have not recovered (=\< grade 1) from adverse events due to agents administered more than 4 weeks earlier (tolerable grade 2 adverse events may be allowed at the discretion of the investigator) * Patients may not be receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MK-2206, lapatinib or other agents used in the study * Patients receiving any medications or substances that are strong or moderate inhibitors or inducers of cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP 450 3A4) are ineligible unless they can be transitioned off this medication prior to study drug initiation * Patients on strong or moderate inhibitors/inducers will become eligible if they discontinue all such medications at least 5 days prior to start of therapy and no further doses are anticipated for the duration of investigational therapy * In order to be considered a contraindicated medication, a patient must be taking the drug systemically and on a regular and scheduled basis; for example, a topical medication taken intermittently need not be stopped * Patients currently taking weak CYP3A4 inducers, and/or inhibitors are eligible * Patients currently taking sensitive substrates with narrow therapeutic indices are ineligible unless: * The medication can be monitored clinically in the opinion of the principal investigator (PI)/Study Chair; monitoring will be performed on a schedule to be determined by the PI/study chair and treating physician (MD); for example, a substrate medication that can be clinically monitored is digoxin * If the medication CANNOT be monitored clinically, they discontinue all such medications at least 5 days prior to start of therapy and no further doses are anticipated for the duration of investigational therapy * Patients with diabetes or at risk for hyperglycemia should not be excluded from trials with MK-2206, but the hyperglycemia should be well controlled before the patient enters the trial * Inadequately controlled diabetes mellitus or hyperglycemia will be defined as: * Hemoglobin A1c \> 8% * Fasting blood glucose over 200 * Diabetes which requires injected insulin * Patients with malabsorption syndrome or other condition that would interfere with intestinal absorption; patients must be able to swallow tablets * A baseline QT interval corrected by Fridericia's formula (QTcF) \> 450 msec (male) or QTcF \> 470 msec (female) will exclude patients from entry on study; medications that may cause QTc interval prolongation should be avoided by patients entering on trial * Patients with a left ventricular ejection function (LVEF) less than 50% or the lower limit of institutional normal are ineligible * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with either MK-2206 or lapatinib * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible