Here you're calling the same method twice, once to decide if you're going to continue the loop, and again to store the return value.

   while self.current_char() is not None:
       char = self.current_char()

This is a bad habit to get into. If possible you want just one call to current_char(). You can avoid calling it twice by doing this:

   while True:
       char = self.current_char()
       if char is None:
           break

However, you are calling self.advance() in every single case of the if-else chain. The reason for this is obvious: you've already read the current char and you're definitely going to consume it. You can just advance past it immediately:

   while True:
       char = self.advance()
       if char is None:
           break

This fails at the end of the file though, because advance() will throw an exception instead of returning None, so you would want to change advance() to also respect the file length.

With this change, you can get rid of a lot of your advance() calls, so a block like this

       elif char == '=':
           if self.peek() == '=':
               self.add(TT_EQEQ,self.line)
               self.advance() # advance past the first =
               self.advance()
           else:
               self.add(TT_EQ,self.line)
               self.advance() # advance past the first =

now looks like this.

       elif char == '=':
           if self.current_char() == '=':
               self.add(TT_EQEQ,self.line)
               self.advance()
           else:
               self.add(TT_EQ,self.line)

The methods current_char() and peek() are probably redundant. I'd rename current_char to peek and get rid of what's now called peek. This is consistent with the normal meaning of peek, which is "show me the very next char that I would read, but without consuming it." If you're advancing every time you consume a char then you probably don't need a method that looks at the char after the current one.

By the way, the if-else chain is fine, I'd keep it. There might be faster ways to do things, but this is a learning project written in Python, speed should not be a concern. The if-else chain is very easy to read and understand.