Using an AVR Programmer to program the TQFP version of ATMEGA328P
Well first of all, check the wiring. Check that the ISP header connects to the pins that it ought to, and check that you have the chip oriented correctly in the socket The USBtiny has a switch to power the target, so make sure that's in the correct position. If all that is correct, there are a couple other things that could cause that error:


The reset pin lacks the normal ESD protection, and so is fairly easy to fry. Ground yourself and use an ESD mat.
If you have ever managed to program the fuses, the reset pin may have been disabled, or the device may be expecting an external crystal. If the reset pin has been disabled, you'll need a special high voltage (12V) programmer to recover it.