Negative presure is actually a serious problem. Modern homes are tight enough that a negative pressure device like a dryer venting has the potential to pull negative pressure on the house and interfere with gravity (convection) exhaust stacks on fuel furnace, water heater, etc. This is why heat exchangers are a thing, and many machines are now direct vented, with double vents, inner for exhaust, outer for combustion air. (Bonus, the exhaust heat warms and dries the combustion air).
Also, it is a shame to throw away perfectly good heat.
The cheap way is to block doors open and set blowers/fans in doorways to pull the heat around the house. In summer, open basement windows. However this is not a reliable solution and requires a lot of awkward fans-in-pathways and manual arrangement.
Other venting plans are up against a basic fact: air is not a very good thermal transfer fluid. Water is ideal, and adding antifreeze doesn't hurt much.
from here on out, when I say water, I mean antifreeze, ideally propylene glycol which is basically edible. A spill will bring vermin rather than kill cats.
Surprisingly little water flow will do a fine job of cooling a PC, even several chips plumbed in series. (Another way of saying this is, heating water is hard).
Water cooling - push the water around
If practicable, use water-cooling on the equipment. This detailed "whole room water cooling" video series goes deep into the nuts and bolts of it. They do a few things I don't agree with, like dingdonging together a bunch of PC radiators instead of just getting an old automobile radiator (with attached thermostats and electric fans) from a junkyard.
But the basics are the same: cooling plates on the hot running chips, pumps in the PCs, a hot/disposal water trunk, and a cool/makeup trunk.
Now in the simple version, you plumb all the heat outside, to an automobile radiator. Keep its built-in fan thermostat and fans. Wire it off a 12V supply, so the thermostat operates a 12v relay which supplies power to the fans. Plumb it so water reaches the thermostat last. That way the fans only run if they're needed, just like in the car.
In the more complex version, you make stop(s) and get to the automotive radiator last, but first, you make a tour around the house and use the water-heat "any which way you can". Now this is very similar to hydronic heating systems used in wealthier homes in the US, and multipurpose boiler" systems in the UK. I'm not an expert in those systems, so I'll explain the general gist. And they may not be in this sequence.
First stop: household heat
You take hot water either to a heat exchanger in the furnace, or baseboard heating around your house, whichever is more practical.
In the normal state, water flow bypasses your furnace or baseboards and goes onto the next thing - that way it doesn't overheat your house. When your house thermostat calls for heat, it stops bypassing and flows througH the furnace/baseboards. You'll need a 2-stage thermostat (or 3-stage if you already have 2-stage). You would want to program this as the first stage. When the house is not warm enough, the thermostat will call first to use this free waste heat. If that works, the thermostat will cycle it on and off (bypassing or not). If that is not enough, it will then call for your normal heat to fire up.
You need a diverter valve. During a call for heat, the valve operates and forces water through the circuit. If the little pumps inside the PC's can't push water around the heating circuit, you may want to have a booster pump turn on with the diverter valve. If the booster pump is a type that tends to block flow when it's off, you may not even need the diverter valve!
Other heat dumps, er, I mean uses
Next stop is the water heater. The bypass valves will be a bit more complex, because if your heater is already 60C, you don't want to run 50C water through it. Or you may have an on-demand heater and let your system "help it out", but in that case don't heat standby water 24x7, as this will breed legionella. (A recently discovered problem that reared its ugly head in the Flint, Michigan water crisis). There's a real science to getting hot water right, without scalding or breeding legionella.
Then you go on to any hydronic floor heating you may have. Now, this is a bit of a problem, because if your servers aren't running for some reason, or if the furnace/baseboards have already taken all the heat out of the water, then no floor heat unless you have an alternative heating method. The other option is to do the floor-heat circuit first before the furnace/baseboards, since the furnace has a plan B, and the floor heat does not.
Finally the outside radiator
Now if everything bypasses, or if the servers are making more heat than the non-bypasssing things can use... Then the heat winds up outside at the aforementioned automobile radiator.
The only other serious concern is condensation. You don't want fresh cooling water arriving inside the PC colder than about 10C, as it risks causing condensation inside the PC. This problem could be reduced by careful ducting around the outside radiator, so it doesn't convection cool very well, and needs the radiator's fans to run to be effective. When the fans cycle off, it shouldn't do much cooling.
If all else fails, have the PCs gravity draw from a tank of water above them, and have the radiator discharge into that tank. That way the tank temperature changes slowly. Then a bypass valve on the radiator when tank temp is below 15C. The tank is to keep the bypass valve from rapidly cycling.