• Histologically-proven systemic anti-light chain amyloidosis (AL) confirmed by positive Congo red staining with green birefringence on polarized light microscopy and evidence of a measurable clonal disease that requires active treatment. An underlying plasma cell disorder can be identified by one of the following: clonal plasma cells in the bone marrow (BM), monoclonal protein in the serum or urine, or abnormal free light chain ratio. For patients who are African-American or males \>= 70 years with isolated cardiac involvement, mass spectrometry must be performed to confirm subtyping

• Presence of t(11;14) by fluorescence in situ hybridization (FISH) on bone marrow biopsy, either confirmed at screening or documented with a prior biopsy

• Patient requires therapy, as determined by the treating physician, following at least one line of treatment (No limit on the number of prior treatments)

• Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of venetoclax in combination with MLN9708 (ixazomib citrate) and dexamethasone in patients \< 18 years of age, children are excluded from this study

• Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)

• Leukocytes \>= 3,000/mcL

• Absolute neutrophil count \>= 1,000/mcL. Screening absolute neutrophil count (ANC) should be independent of granulocyte- and granulocyte/macrophage colony stimulating factor (G-CSF and GM-CSF) support for at least 1 week and of pegylated G-CSF for at least 2 weeks

• Platelets \>= 75,000/mcL. Platelet transfusions to help patients meet eligibility criteria are not allowed within 2 weeks before study enrollment

• Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN)

• Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN

• Creatinine Calculated clearance \>= 15 mL/min using Cockcroft-Gault equation

• Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial

• For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated

• Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load

• Patients with a prior or concurrent 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

• AL Amyloidosis Cardiac Risk stage I, II or IIIa disease based on the 2013 European Modification of the 2004 Standard Mayo Clinic Staging in patients with advanced cardiac involvement (Dispenzieri et al., 2004; Wechalekar et al., 2013)

‣ Staging system defined by: NT-proBNP cut off of \< 332 pg/mL and troponin I cut-off of \< 0.10 ng/mL (in the absence of troponin T, troponin I \>= 0.1 ng/mL can be used) as thresholds for stages I, II and III; NT-proBNP \< 8500 pg/ml for stage IIIa

⁃ Stage I, both under threshold;

• Stage I: Zero markers above threshold: NT-proBNP \< 332 ng/L AND troponin T (TnT) =\< 0.035 ng/mL; NT-proBNP \< 332 ng/L AND TnI =\< 0.1 ng/mL

⁃ Stage II, either troponin or NT-proBNP (but not both) over threshold;

• Stage II: One marker above threshold: NT-proBNP \>= 332 ng/L OR TnT \>= 0.035 ng/mL; NT-proBNP \>= 332 ng/L OR TnI \>= 0.1 ng/mL

⁃ Stage III, both over threshold;

⁃ Stage IIIa, both over threshold but NT-proBNP =\< 8500 pg/ml

• Stage IIIa: Two markers above threshold: NT-proBNP \>= 332 ng/L BUT =\< 8,500 ng/L AND TnT \>= 0.035 ng/mL; NT-proBNP \>= 332 ng/L BUT =\< 8,500 ng/L AND TnI \>= 0.1 ng/mL

∙ Stage IIIb: Two markers above threshold: NT-proBNP \> 8,500 ng/L AND TnT \>= 0.035 ng/mL; NT-proBNP \> 8,500 ng/L AND TnI \>= 0.1 ng/mL

• Life expectancy \>= 3 months

• Plasma cell burden =\< 60%

• Absence of bone lesions and other end organ disease consistent with multiple myeloma (patients with plasma cell burden between 10 and 60% without end organ disease can be included)

• Measurable disease of AL amyloidosis as defined by at least one of the following: 1) serum or urine monoclonal protein \>= 500 mg/dL by protein electrophoresis, or 2) serum free light chain \>= 20 mg/L with an abnormal kappa:lambda ratio or the difference between involved and uninvolved free light chains (dFLC) \>= 20 mg/L

• It is not known what effects MLN9708 (ixazomib citrate), venetoclax, and dexamethasone have on human pregnancy or development of the embryo or fetus. Therefore, female patients participating in this study should avoid becoming pregnant, and male patients should avoid impregnating a female partner. Nonsterilized female patients of reproductive age group and male patients should use effective methods of contraception through defined periods during and after study treatment as specified below.

‣ Female patients must meet 1 of the following:

• Postmenopausal for at least 1 year before the screening visit, or

∙ Surgically sterile, or

∙ If they are of childbearing potential, agree to practice 2 effective methods of contraception from the time of signing of the informed consent form through 90 days after the last dose of study drug, or

∙ Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods\] and withdrawal are not acceptable methods of contraception)

⁃ Male patients, even if surgically sterilized (i.e., status postvasectomy) must agree to 1 of the following:

• Practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug, or

∙ Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods for the female partner\] and withdrawal are not acceptable methods of contraception)

• Left ventricular ejection fraction \>= 35% by echocardiogram.

• Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible