Why is this a bad MOSFET/Driver for a Tesla Coil, what does one look for in a good one?
Here is an approach to quantitatively screen for a MOSFET most likely to match requirements. Equations used here will be based on those from the thread "micro, MOSFET, and DC motors", but will be rearranged and reformulated to better reflect MOSFET datasheet parameters. 

Basic Static Criteria: 


\$V_{\text{DS}}\$ ~ \$1.5 V_{\text{s-max}}\$ : 
\$V_{\text{DS}}\$ shouldn't be less, but also shouldn't be much higher than 1.5 times supplied voltage. 
\$V_{\text{Drv-min}}\$  > 2\$V_{\text{th-max}}\$:
If peak drive voltage is less, the FET channel conduction will not be fully enhanced. 
\$\text{$\Delta $T}_{J-A}\$ &lt; 50C : In the approach that will be shown, temperature rise and part thermal resistance will be used to set overall power criteria. The aim is to keep FET junction temperature less than 120C, which a temperature rise of 50C will do even if the ambient temperature is 70C. For a more reasonable ambient temperature of 50C a \$\text{$\Delta $T}_{J-A}\$ of 50C, of course, results in a junction temperature of 100C, which is what we'll use in the selection criteria. 


Total power dissipated in the FET will be temperature rise divided by thermal resistance: 

\$P_T\$  = \$\frac{\text{$\Delta $T}}{\Theta _{\text{JA}}}\$= \$P_{\text{cond}}\$ + \$P_{\text{sw}}\$ =  \$R_{\text{ds}}\$ DC \$I_d^2\$ + \$ I_d V_s F_{\text{PWM}} \tau\$ 

where DC = duty cycle and FET switching time \$\tau\$ = \$\frac{2 R_g Q_{\text{mp}}}{V_{\text{drv}}}\$, 

I will state, without proof, that the lowest power will be attained by having \$P_{\text{sw}}\$ = \$P_{\text{cond}}\$. Therefore in the following equations, both \$P_{\text{cond}}\$ and \$P_{\text{sw}}\$ will be replaced by \$\frac{\text{$\Delta $T}}{2 \Theta _{\text{JA}}}\$, or 1/2 \$P_T\$.

Dynamic Selection Criteria:

Then selection equations for \$R_{\text{ds}}\$ and \$Q_{\text{mp}}\$ can be written as: 

\$R_{\text{ds}}\$ = \$\frac{ \text{$\Delta $T}}{3 I_d^2 \text{ DC } \Theta _{\text{JA}}}\$ : Recall that \$R_{\text{ds}}\$ here is scaled by 2/3 to account for junction temperature of 100C and positive temp coefficient of \$R_{\text{ds}}\$

\$Q_{\text{mp}}\$ = \$\frac{3 \text{$\Delta $T} V_{\text{drv}}}{4 I_d F_{\text{PWM}} R_g \Theta _{\text{JA}} V_s}\$

Example:

For this case the defining parameters are:


\$V_s\$ = 170V 
\$F_{\text{PWM}}\$ = 150kHz
\$I_d\$ = 3Amperes 
\$V_{\text{drv}}\$ = 10V
\$\Theta _{\text{JA}} \$ = 62C/W (for TO-220 or TO-263)
\$R_g\$ = 20 Ohms
DC = 0.5 


These yield search parameters: 


\$V_{\text{DS}}\$ = 250V
\$V_{\text{th-max}}\$ &lt; 5V
\$R_{\text{ds}}\$ =\$\frac{\text{50C}}{\text{(3)(9)(0.5)(62C/W)}}\$ = 59.7mOhms 
\$Q_{\text{mp}}\$ = \$\frac{\text{3 (50C)(10V)}}{\text{4 (62C/W)(3A)(150kHz)(10Ohm)(170V)}}\$ = 1.28nCoul


Here is the search result from Digikey 

The best match was an IPP600N25N3 , which had a \$Q_{\text{mp}}\$ of 3nCoul, so in order to meet power dissipation requirements either \$F_{\text{PWM}}\$ would have to be lowered to about 50kHz, or a heat sink would be needed to lower thermal resistance.