Here's an example from the Murphy's grammar textbook:
"I have to get up early tomorrow. I have a lot to do."
So my question is: shouldn't it be "I WILL HAVE to get up early tomorrow." because of "tomorrow"?
I would suggest you are probably currently in the condition of having a lot to do tomorrow as the tasks have likely already been allocated. However, saying you "will have a lot to do" implies you aren't already in the condition of having a lot to do tomorrow, which probably isn't the case.
You might say "I have to get up early tomorrow. I will have a lot to do." if were referring to tasks you know will be allocated, but which haven't been allotted just yet.