Does anyone know of a place to hire motorbikes in Thailand, say Bangkok or Chiang Mai, and ride them into Laos and back? The trip will be self-guided and should take no longer than two weeks.
Yes, it is possible to hire motorbikes in Bangkok and Chiang Mai. I haven't done this myself but some friends did. It costs about 200-800 baht per day to hire one depending on the type of motorcycle you get. It might make sense to buy one from an expat or a another traveller, then sell it when (if?) you go back to Thailand. You will definitely need to have an IDP to drive around in Bangkok as police keep a watch out for tourists.
The problem I foresee with your plan of driving to Laos could be that many motorcycle rental places in Thailand retain your passport. You could try convincing them to leave just a copy of your passport but for long-term hires they may not agree. You also need to have paperwork to cross the Thai-Lao border on bike.
Those are the basics. GT Rider is a good place to start your research on the exact documentation you need to cross the border, the kinds of motorcycles usually available for hire, maps, et al.
I hired a scooter in Chiang Mai for 200 baht a day. However, an accident there might cost you about 4000-5000 baht. You would be very lucky if the damage is minor and you are quoted a damage cost below 4000 baht. No license is needed, just a passport will do. They will keep your passport when you rent the bike.
The rented motorbike does not usually have enough milage to bring you from Bangkok/Chiang Mai to Laos. Furthermore, I highly recommend against travelling to Laos as the roads in the Laos section of your journey may be bumpy and not well-suited for your rented motorbike.
Hiring a motorbike in Thailand isn't a problem. From THB 200 per day of THB 2,000 per month you'll 'have' one.
As another poster stated that a rental-company will retain your passport: that's not allowed. They are allowed to make a copy of your passport incl. visa page and Immigration Departure Card.
The problem is 'entering Laos'; most probably no rental-company will allow you to take the rental-bike into Laos. Temporary import of a vehicle into Laos, Cambodia is only possible if you have the original green/blue book on hand OR if you have a written permission from the owner of the vehicle (there's a special form for that), but the rental company will be reluctant/refusing its cooperation.