Slayer exciter (mini Tesla coil) not working with IRFZ44N
This is "not quite" the definitive Slayer Exciter SE EE reference BUT it should serve as a useful guide to Slayer Seekers. If it seems useful I'll add to it. 

A "Slayer Exciter" implementation of a Tesla Coil is a somewhat hit &amp; miss affair at the best of times. Some work well, or sometimes work well, or ... . Many are not designed or poorly designed. And as with anything found on the internet - some will never have worked for the maker, no matter how convincing their video or text may (or may not) be, and no amount of copying correctly will make your version of their fraudulent material work.

Your example is suspect in using a a MOSFET (very unusual, not impossible) and lacking the usual base (or gate) to ground reverse connected diode, which is an essential part of most such circuits.

His long winded and unnecessarily detailsed warnings about destroying the MOSFET gate are correct but his chosen component values are dangerously bizarre given his warnings. His suggested range of operating voltages are unnecessarily high.
All told, while his circuit MAY in fact have worked for him, it's not by design, and I'd suggest you look through a few dozen of the hundreds available on web and choose one more liable to be a good one. 

All that said:

"Dotting" or relative winding direction and plarity of the two inductors is CRUCIAL. In my incomplete skim of his video he did not seem to mention this and did not show it on his diagram - an utterly inexcusable omission.
If both coil are wound in the same direction on the former then the bottom of the low voltage one should connect to the transistor collector / drain. This is sort-of-if-you-are-lucky implied by his diagram. That's not good enough.
Reversing the primary polarity may make the difference.

Operation from 12V should be possible. This may even save your FET gate from dying.
A few turns on the primary may help. Or not.
A diode from gate to ground (cathode / bar end to gate) may help. 

Here is an excellent SE EE Q&amp;A re Slayer Exciters

Here are 46 SEEE Slayer Exciter posts in descending order of votes.

Hare are agh agh agh!!! 100's of the brutes. 

Screen capture from his video.
Dots by me.
Build another version.