Can I supply a device rated for 1A with 15A power supply?
Think about it.

You have an electrical supply to your house. How many amps can it supply. My main fuse is rated at 63 A.

You have a light bulb. It's rated at 20 W. How many amps will it draw. Answer: \$ I = \frac {P}{V} \$ so that's about 0.1 A in 230 V land or 0.2 A in 110 V land.


  I'm talking about a big difference between output current of the power supply and rated current of the device here.


Me too.


  Would it burn up?


No. As in the case of the light bulb, the load will draw the current it requires and that's all.

You may want to consider limiting the current to a safe value in the event of a fault. Add a 2 A fuse in-line and you should be fine.