Please post more acurat questions. There are multiple Libraries available. Which one are you using?
If you are using my wrapper then read the following answer: (ignore it if you are using another library)
The bot is, as its name is suggesting, designed to run all the time. When the bot is running all messages coming in are raised in an Event. I never heard of any Problems.
If the bot is not running and someone sends a message to him; i´m not sure if they are suspect to survive in the Telegram network. You probably loose them if your bot is not running.