"Remember to do something on time" 的英文表达是 "remember to do something in a timely manner" 或者 "remember to do something on schedule".

这个表达的来源是由 "remember" 和 "timely" 或者 "schedule" 这两个词组合而成的。"Remember" 意思是 "记得"，"timely" 意思是 "及时的"，"schedule" 意思是 "时间表" 或者 "安排".

下面是一些例句：

Remember to submit your assignment on time. (记得按时提交你的作业。)
Please remember to arrive at the meeting on schedule. (请记得按时到会。)
I always remember to take my medication in a timely manner. (我总是记得及时服用药物。)
We need to remember to renew our membership on time. (我们需要记得按时续费会员。)