Most (native) American English users say:
I'm as tall as him.
Since the word him is not functioning as an object in the above sentence, it is grammatically incorrect. The correct usage is:
I'm as tall as he.
Since the grammatical usage is the less common, it tends to sound odd to native speakers. My advice is to use the correct pronoun, but don't leave it dangling.
I'm as tall as he is.
The ants worked on their anthill with as much dedication as he has.
I drive faster than he does.