I would worry about the shop vac overheating but that is apparently not a problem. Would think that no cooling air moving thru the vac would build heat?
Air through the vac chamber isn't the problem, meaning the restrictions through the transmission/hydraulic reservoir isn't likely to harm anything about your vacuum. The air through the vacuum reservoir doesn't go through the motor unless you have a very poorly built vacuum. That's why most of them have vents on the top, and a separate discharge for blowing air. There are also vents down near the lid, where air is drawn in and blown through motor. The float/check valve is to keep liquid out of the fan which could affect the bearings as well as throw a rather high speed fan out of balance. THAT's the problem, but I would leave the foam primary filter on just to give that little dab of insurance. It's what I do when I use my cheap vacuum for picking up water. There's basically a tornado inside that reservoir, and a lotta splashing going on once there's more than about an inch of liquid in it. First time I saw water spit out the discharge, and the vacuum started walking around by itself, I was convinced the primary foam filter would remedy that. It did. There's a diverter on the inside of the vacuum side to make sure all liquids go down away from the fan. It isn't an oil tight or water tight seal, but the last thing you want is damp/dirty air from the inside of the vacuum going through the motor. It won't last. If you see moisture around the motor cooling vents, you've likely spun the bearing for the fan in it's usually plastic mount, and it's time for a new vacuum.
If you have other vent holes on the tractor (case vent), including when you remove the suction filter (the worst one) all it can do is draw air into the transmission. I would be extra cautious about making sure the area around the filters is clean to make sure I didn't pull any dirt/moisture into the oil I'm trying to save. If it isn't enough vacuum to hold back the oil, then pinch off the case vent with a clothes pin or small clamp.
As for burning up or overheating, probably not going to happen. In the first paragraph, note I said the motor doesn't need the suction fan for cooling. The fan (drive end) bearings are sealed high speed bearings. Good to go unless your vacuum is as ancient as mine. Never need lubricating, which is a good thing because they're sealed anyway. Mine howls a little when I turn it off. That's your sign you're gonna need a new one soon. The motor has MORE load on it when the fan is actually moving air. The more air flow, the heavier the motor load, the hotter the motor gets. Shut the air flow off, the motor current decreases, the motor speeds up (supplying more cooling air to itself) and will likely run that way for ever. If the motor starts overheating, it's likely because dust/debris has fouled the cooling air path. It was gonna overheat anyway. Again, your sign that it's time for a new vacuum, or if you're mechanically and electrically agile, time to clean it really good. You're not likely to find parts for cheap vacuums, though, and if you do, it'll probably be cheaper to replace the vacuum.
But, considering the time it takes to get things cleaned up, the vacuum attached, and someone to hold it, I can have the two drain plugs out and the fluid drained into a CLEAN catch pan so I can have a look and make an educated choice about replacing the fluid or not. No drips. no drops. I'm not saying it's the best way, because I have to funnel it back in if I'm re-using it. It's just the way I prefer.
