Why is my Tooth So Sensitive Still After 3 Weeks of Filling?

The deeper and more extensive the filling the less natural protection there is for the nerve of the tooth.  This could lead to both reversible nerve sensitivity as well as irreversible nerve sensitivity.  Many times it can take quite a while for the nerve to repair itself.  I would suggest seeing your dentist to rule out any high bite issues with the recent filling along with a new xray.    If the symptoms get worse or happen more frequently the tooth may need a root canal procedure.  If it stays the same or gets better over time I would hold off on having a root canal procedure.
Good luck.

Sensitivity to cold after 3 weeks can be concerning.  Go back to your dentist to have him take a look at it.  If the filling is large, then it may be deep near or even in the nerve/pulp.  Hopefully it's just a minor bite issue where the filling is too high.  Please let us know what the result is.

Brajesh, thanks for you question. There are several reasons why your tooth can be sensitive to cold after a filling was done.
One, cavity was located very close to the nerve inside the tooth. Another reason, when you are numb at the time of filling placement, bite might seem to be fine, but in reality it might be just a smutch high or ,third, it might be interfering ,when you move your teeth all the way left or right. Also you might be a grinder. If so, additional pressure from that can create sensetivity.
My advise is to see a dentist, who placed that filling asap and let him diagnose the problem.

Sometimes teeth stay sensitive after getting a new filling. This happen often after treating teeth with deep cavities. A previously infected tooth pulp many not recover with treatment. Another common reason is a high bite on the new filling. Consult with your dentist about the sensitivity. Good luck. 

The closer proximity you get to the nerve when doing a procedure...the less insulation there is for the nerve (bite and temperature).   Sometimes, it just takes a little while for the reparative functions of the tooth to kick in.  I always tell patients that as long as they think it's getting slightly, slightly better each day to hold off.  In the absence of your bite being off, typically this reversible pulpitis will calm down.