Most places will accept a photocopy along with a security deposit - same with cars. Just need to say you need to do a visa run and need the passport. I live here in Thailand and have never had a problermproblem renting anything from Honda Dreams to trucks. Just make sure you check it before you ride (as there are plenty of damage-scams about) and also pay more for the insurance (and read it - it will/should be in English).
In Chiang Mai I always go to POP to hire bikes (sometimes cars too - but there are better places for them) - there are several POP's in the city (it's like a franchise), but I always use the one next to Spotlight (gogo bar on the moat) just before Thaipae Gate (down a bit from Miguels burgers, before you get to Loi Kroh where the bars are mostly situated). They do Dreams, D-Trackers, ERN6's and Kwak8's. 800's cost a cool 1,000 per day (local prices - may be higher for tourists) - the price of a car rental. Usually deposit is 5,000 Baht (they staple it to the photocopy and the paperwork that shows previous damage etc that you sign). Pop vehicles are fully insured - they advertise the fact. And no, other than being a time-to-time customer, I am not affiliated. When my car was off the road (awaiting a new engine which took several months) I had both cars and bikes off them of varying terms - there's a daily, weekly and monthly rate.
//Edit: Just to add: On the photocopy, write clearly "For rental purposes only" and sign it - there ahvehave been rare occurancesoccurrences where these are have been used for other purposes (such as getting loanshark loans)! Possibly an urban myth, but advisable anyway.
