How to connect a 24v air purifier to hvac

Judging by the Honeywell R7184A Controller manual, you have one of these:

You described it as terminal 4 but the diagram just shows two terminals labelled "T", but that is fine:

I found a manual for an EnviraCom device which shows terminals 2 and 3 are 24vac power:

This means we have the right connections, and according to the R7184 manual:

EnviraCOM™ Current Available: 150 mA

So the liming factor here is simply the current available. I can't find any specs at all for the thermostat you posted, but so long as it needs 150 mA or less (at 24 Vac) then it should work. You would make the following connections:

Burner             Thermostat     Desc
Terminal 4 T       W              Heating call
Terminal 3 T       R or Rh        24Vac
Terminal 2         C              24Vac "Common"

Note: your current wiring may not have W and R connected correctly, because with the two-wire system it doesn't matter. Now that you need a C wire, it is important to have R connected to constant power. If wrong, your thermostat simply won't get power.

If your thermostat draws more than 150 mA, you're going to run into various strange problems that may range from occasional glitches to your burner not working at all, and I'd highly advise against doing this.

If you do need more than 150mA, normally you could upgrade the transformer -- but in this case, it's all an integrated solid-state unit. I'm actually not sure you could wire this up without damaging the burner controller. The safest thing would be to use a separate circuit with a relay, but that is far beyond the original scope so I won't post how do to that unless necessary.

Yes, tapping along with the red wire under the (C)OM is where it goes. If you'll notice, your (Y) is already double taped.