A Study of JNJ-67571244 in Participants With Relapsed or Refractory Acute Myeloid Leukemia (AML) or Myelodysplastic Syndrome (MDS)

Inclusion Criteria:

* A diagnosis of:

  a) Acute Myeloid Leukemia (AML) according to the World Health Organization 2008 criteria with relapsed or refractory disease and ineligible for or have exhausted standard therapeutic options b) high-risk or very high-risk Myelodysplastic Syndrome (MDS) according to International Prognostic Scoring System (IPSS-R) and relapsed or refractory after at least 1 course of hypomethylating therapy and ineligible for or have exhausted standard therapeutic options per investigator discretion should be included
* Eastern Cooperative Oncology Group (ECOG) performance status grade of 0 or 1
* Women of childbearing potential must have a negative pregnancy test at screening and prior to the first dose of study drug using a highly sensitive pregnancy test (either serum or urine beta human chorionic gonadotropin \[beta-hCG\])
* Chemistry laboratory parameters within the following range during screening:

  a) aspartate transaminase (AST) and alanine aminotransferase (ALT) less than or equal to (\<=) 3\* upper limit of normal (ULN), b) Total bilirubin \<=1.5\*ULN; participants with congenital bilirubinemia, such as Gilbert's syndrome, may enroll if conjugated bilirubin is within normal range, c) Creatinine clearance calculated or measured creatinine clearance greater than or equal to (\>=) 30 milliliters per minute (mL/min)
* Before the first dose of study drug:

  1. Women of childbearing potential and fertile men who are sexually active must agree to use a highly effective method of contraception (less than \[\<\] 1 percent {%} year failure rate from the time of signing the informed consent form \[ICF\]) during the study and for 90 days after the last dose of study drug. Contraception must be consistent with local regulations regarding the use of birth control methods for participants participating in clinical trials. 1) Participant must agree to practice a highly effective method of contraception (failure rate of \<1% per year when used consistently and correctly). Examples of highly effective contraceptives include: a) user-independent methods: implantable progestogen-only hormone contraception associated with inhibition of ovulation; intrauterine device; intrauterine hormone-releasing system; vasectomized partner; b) user-dependent methods: combined (estrogen- and progestogen-containing) hormonal contraception associated with inhibition of ovulation: oral, intravaginal, and transdermal; progestogen-only hormone contraception associated with inhibition of ovulation: oral and injectable, c) In addition to the highly effective method of contraception, a man: 1) Who is sexually active with a woman of childbearing potential must agree to use a barrier method of contraception (for example, condom with spermicidal foam/gel/film/cream/suppository), 2) Who is sexually active with a woman who is pregnant must use a condom, c) Women and men must agree not to donate eggs (ova, oocytes) or sperm, respectively, during the study and for 90 days after the last dose of study drug

Exclusion Criteria:

* Willing and able to undergo allogenic stem cell transplant \<=6 months before the first dose of study drug, has evidence of graft versus host disease, or requires immunosuppressant therapy (exception: daily doses less than 10 milligram (mg) prednisone or equivalent are allowed for adrenal replacement)
* For Part 1 only, prior treatment with CD33 targeting therapy targeting T-cell redirection (for example, CD-3 redirection technology or chimeric antigen receptor \[CAR\]-T-cell therapy)
* For Part 1 only, prior Grade 3 cytokine release syndrome (CRS) related to any T-cell redirection (for example, CD-3 redirection technology or CAR-T cell therapy)
* Prior treatment with a checkpoint inhibitor such that the first dose of JNJ-67571244 would occur within less than 5 half-lives. Prior treatment with chemotherapy, targeted therapy, immunotherapy, radiotherapy, or treatment with an investigational anticancer agent, an investigational drug (including investigational vaccines), within 2 weeks prior to the first dose or at least 4 half-lives, whichever is less, or currently receiving investigational therapy in a clinical trial. Hydroxyurea may be used
* Toxicities (except for alopecia, peripheral neuropathy, thrombocytopenia) from previous anticancer therapies that have not resolved to baseline levels or to Grade 1 or less