Is it considered unprofessional to address a business envelope by hand?

Sometimes it's just faster. I have neat handwriting and think it looks more personal.

If you want a professional answer, No. But a lot of small, privately owned companies seem to do it.
I'd say if you were sending a resume....no.
If you are sending a proposal to another business....No
If you are sending an invoice.....No.

It puts an outward impression on doing business to business.
It's like saying is it ok to wear jeans at a wedding or funeral....No!

I think it's fine, especially nowadays when there are rarely typewriters around.

It depends on the nature of the coorespondence. If its a thank you note, then there is really no problem. If you are sending a resume, better type the address out. If in doubt, type it.

If you're sending correspondence from a business to another business it's better to have it typed. It may take time but it is more businesslike. You said that the handwritten address is more personal and that it is.

some businesses do it and it does not deduct the professionality of it if you hand-write it and it IS NEAT!

It depends on what field you work in. But I used to work at a high-end insurance company, and when I'd deliver mail, the handwritten addresses were always the first to get opened. But again, it depends on the field.