Based on prices we saw around Chiang Mai during our 5 weeks here, this is what you can expect to spend per day in Chiang Mai, as of February, 2014 :

breakfast ฿ 25 - 100+ You can get something from a street vendor or at 7-11 for under 30 baht, but if you opt to go to a restaurant for Western breakfast, expect to spend a lot more.
lunch ฿ 30 - 100+ There are a lot of dishes available around 50 baht at a restaurant; for example, stir-fried meat and vegetables on rice. It's definitely possible to spend even less (30 baht) if you eat from a street vendor or a local place. Things are cheaper outside the Old City.
dinner ฿ 30 - 250+ Like lunch, if you eat something at a local place, you can easily get a filling meal (something over rice or fried noodles) for under 50 baht. If you want Western food or prefer to eat in a nicer, more touristy restaurant, of course you need to pay more — a nice meal at a decent place for 2 should cost under 400 baht, including drinks.
snacks & drinks ฿ 5 - 150+ It costs 6 baht for a 600 mL bottle of water & 17 baht for a 1.5 L bottle. You can also refill your bottles from water machines for just 1 baht. Juices and smoothies range from 25-60 baht. Drinks from the convenient stores or supermarkets are cheaper of course. Beer is at least 50+ baht. Cocktails, 70-100+ baht. Chips, yogurt, chocolate, and other snacks are usually 10-30 baht each.
Transportation ฿ 0 - 500+ It's pretty easy to get around to the amazing temples in and out of the Old City on foot. If you want to rent a bicycle, you can easily get one for 50 baht per day and it's flat so easy to cycle although the traffic can be pretty bad. Motorcycle/scooter rentals are 200-800+ baht. Taxis are usually a flat rate of 120-150 baht within the city. Tuk tuks (motorcycle taxi) and songthaew (shared pickup truck taxi) start at 50 baht, but you need to negotiate your price.
Accommodation ฿ 200 - 1000+ You can get a pretty nice room at a decent hotel with a pool for under 700 baht. If you're staying for longer than 2 weeks, look into monthly rentals. It will make your stay a lot cheaper even if you're not actually staying the whole month!
Other Expenses ฿ 25 - 500+ Laundry (wash, dry & fold) is 40+ baht per kg. Coin-operated washing machines are 30 baht per wash. Internet cafes are 10-15 baht an hour. Insect repellent is 200-300 baht per bottle. Massages start at 180 baht per hour. If you want to join a tour, expect to pay 600-2500+ baht for a day trip. Factor in shopping money as well, but clothes and trinkets are relatively cheap. You can get a t-shirt for under 100 baht.
TOTAL ฿ 315 - 2600+

฿ 215 - 1850+
/ couple

About USD $ 10 - 80+ if you're travelling alone.

About USD $ 7 - 60+ each for a couple due to shared transportation & accommodation costs

If you do everything on the cheap and don't spend money on any extra stuff, it's possible to live on the lowest end of the budget without feeling deprived, especially if you're staying here long-term; however, a more realistic budget so you can spend your time in Chiang Mai comfortably is more like 500-1000 baht a day (depending on your accommodation), all inclusive, plus any activities you may choose to do. With a mid-range lifestyle, splurging sometimes and saving sometimes (not doing touristy things everyday), we spent just under 800 baht each per day, including everything.

In other touristy places in Thailand, however, expect to spend at least 1000-1500+ baht per day.