I replaced the 240vac to 24vac transformer with exact replacement part on my luxaire air handler.. I checked 24vac on transformer before I connected to control card and it was good.. I connected leads to control board and after about 30seconds the transformer burnt up again. The primary coils reads open on the transformer (I didn't check resistance readings of transformer before I installed it). Circuit breakers on unit don't trip or 5a fuse on control board doesn't blow.

You need to buy a transformer with a built in reset switch. It is a breaker built in. Just tell them the voltage and VA rating you want and tell them one with a reset button on it. They cost more but it sounds like your guessing and it will save you in this case some money until you get the wiring right. The unit should have a schematic on or in it. Trace each wire and make sure it is where it shows on the schematic. Some one like yourself probably tried to fix or replace something and you have to find what is not done correctly now. HVAC techs have to go to school for a long time to learn this trade plus 5 years apprenticeship even then you will be learning but you have the technical ability to figure it out. When you get into controls and its voltage you need to be experienced in it or you will in most cases be paying guys like me even more to repair the damage you done and it ends up costing you more money then what it would have to begin with. If you replaced the t-stat it is probably not right, Check the t-stat wires out side also. Dogs, kids, weather break and or decay and short out.

Sounds like you got a bad short in the control board. You can add a 5 amp automotive fuse inline on the 24v side of the transformer using just two spade conectors and the fuse. This will keep the transformer from blowing.

Questions to get to the problem. Do you have a heatpump? After you install a new transformer with a inline fuse as T-Top suggested,, turn off the breaker to heatpump and disconnect 1 (one) wire (smaller tstat wire) that runs to the heatpump,, does it still blow the inline fuse? I`m trying to rule out the possibility of the contactor in your heatpump being shorted and blowing the transformer fuse. Let us know.

I called an HVAC tech so I didn't do any other possible damage.. The tech replaced the transformer with a bigger one that had a built-in circuit breaker.. then by putting on the wires one by one and checking for a spark he found that the red wire on the outside unit had shorted to the metal cover.. all totaled $227.
Thanks for your help.

Thanks for the report back, so very seldom anyone gives a return report of a repair,, glad you have heat and the repair is made