Filttering noise out from high frequency 1MHz oscilator (Teslacoil driver)
I don't see a bypass cap on U2 -- or any of the chips, for that matter.  You should put 100nF caps on all three, positioned as close as you reasonably can to the power supply leads.

If that doesn't work, figure out the current draw of U2 and put a series resistor between the +5V supply and the chip, sized to drop no more than 100mV.