Call your surgeon. Anytime you have an issue after having a surgery it is your responsibility to inform your surgeon. Don't be afraid.
Yes. During surgery, it is common for small nerves to be stretched or even cut. A temporary burning sensation can result while the nerve heals. In the early days/weeks after surgery, it is important to keep the limb elevated to reduce swelling, which can exacerbate pain. Anti-inflammatory medication can also help (clear with doctor (s) first). Prolonged periods of burning should be reported.
Possibly. This could be normal post-operative pain, but f you have any concern of worsening or persistence of the sensation you should discuss it with your hand surgeon.