After screening, participants are assigned to either the dose escalation or dose confirmation phase. The dose escalation phase consists of 2 cohorts and will evaluate available PK and safety data from the first 6 participants of each cohort, including dose limiting toxicities (DLTs). The purpose of the dose confirmation phase is to gather additional safety, tolerability, PK, and preliminary efficacy data of MK-1308 in combination with pembrolizumab. The 4 arms of the dose confirmation phase will include advanced/metastatic non-small cell lung cancer (NSCLC) and second line advanced/metastatic small cell lung cancer (SCLC). In participants who have initial evidence of radiological progressive disease (PD) by Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1, it will be at the discretion of the investigator whether to continue a participant on study treatment until repeat imaging is obtained.