The flight school I am going to has a Cessna 152 that they claim is IFR capable, the aircraft is not equipped with pitot heat. Am I able to take my instrument check ride in that aircraft..?
Yes. 14 CFR 91.205 describes the minimum equipment required for IFR flight. Pitot heat is not included. If the examiner asks you if having pitot heat a good idea, I would say “yes”. If you encounter icing, the pitot tube will most likely be one of the first devices to malfunction due to it's position facing the relative wind and the small diameter of the hole being easily blocked by ice. Without pitot heat to clear the ice, the airspeed indicator is misleading and can cause you to get into an inadvertent stall/spin situation.
A good IFR training aircraft doesn't necessarily imply it will perform in actual IFR. A 152 would be great for training. But, I don't know many pilots that would set off in actual IFR conditions in one. Definitely not without pitot heat…
I've done a good bit of IFR flying in C-150/152's over the years (actual as well as simulated ), including training a few folks for their IR. It's perfectly OK as long as you observe the performance limitations of the aircraft with regard to weight and environment. And Andrew is right about pitot heat being merely because desireable rather than legally required for IFR flight per 91.205 – just stay away from clouds and precipitation when the temperature is below 40F/5C.
The problem with instrument training (or any other training) in a C-150/152 is that the useful load that was adequate for training 65 years ago when the plane was designed and people averaged 170 lb is no longer adequate today when (according to the CDC) people average 200 lb. The result is a reduction of up to 10 gallons in the fuel you can carry, which is nearly half the plane's usable fuel. When I was running a university aviation program, our contracted flight school had four such planes. We had to limit fuel to 16 gallons to make it work, and that meant limiting flight durations and dip-sticking after every flight before precision fueling. So unless you and the instructor your school is providing and the DPE who will be giving the practical test are all lightweights like me, it's not going to work well.
Beyond that, it's mainly a matter of the avionics. The plane must have VHF nav and comm including a glide slope since you must demonstrate a precision approach on the checkride. I've trained people with as little as a single KX-170 with GS (plus transponder), although ADS-B-out is also widely required today. If there is no ILS nearby, you'll need a WAAS GPS so you can substitute an RNAV(GPS) LPV approach for the ILS. Make sure that 152 is so equipped because many aren't.