Why does my new water heater spill exhaust when the furnace kicks on?

A short puff of spillage the instant the furnace lights is one thing, but a minute or two of fumes rolling out of the draft hood is not normal, window or no window. What you're describing sounds like the classic shared flue cold-start problem. When the furnace burner kicks on, it dumps a big slug of hot exhaust into a flue that's still cold from sitting idle. Until that whole pipe warms up, the flue can't handle both appliances at once, and some of the furnace exhaust takes the easy way out, wich is backwards through the water heater. Once everything heats through, draft establishes and it pulls fine, which matches exactly what you're seeing. As for why it started with the new tank, the height is the first thing I'd check. If the new heater is taller than the old one, the draft hood sits higher up the flue and closer to where the furnace ties in, which makes it way easier for the furnace to pressurize that section. Same deal if the new connector pipe got routed differently or has less rise off the top of the tank. Dig up the spec sheet or just measure old versus new if you remember the old one, because a few inches can matter on a shared vent.

The window trick helping a little tells me the utility room might be on the tight side for combustion air too, especially if there's a louvered door or the room is small. But that's a contributing factor, not the whole story. A shared flue has to be sized and pitched for both appliances running together, and that's on the installer to verify against the tank and furnace manuals, not on you to babysit with an open window every winter. I'd call the installer back without hesitation. This isn't you overthinking it. They swapped a component in a vented system and the venting behavior changed, so it's their job to come back and confirm the flue sizing, the rise off the tank, and the tie-in heights. Any decent installer will do that visit for free. If they brush you off, get whoever you trust to look at it, because spillage into a mechanical room every furnace cycle is not something to live with.

Not the same setup but I had our furnace swapped two winters ago and the water heater started spilling a bit after that, just the mirror image of your problem. Turned out the new furnace connector sat a bit differently where it entered the flue and the water heater paid for it. One thing the tech did that might help you describe it to the installer: he held a lit incense stick near the draft hood while everything fired up and watched where the smoke went. Made it easy to see the spillage on video instead of trying to explain it. Did the plumber say anything at all about the vent when he left, or did he just swap the tank and go?

Chiming in late here but something similar bit me about fifteen years back. Swapped tanks, spent a week blaming the furnace, and it turned out the new vent connector never got screwed back into the flue collar, just sitting loose in there. Two sheet metal screws and a proper seat fixed it in ten minutes. Before you get deep into flue sizing theories, eyeball what the installer actualy did up top. Old trick from before the fancy testers came along: hold a strip of tissue paper at the hood with the furnace running. If it blows outward instead of getting pulled in, you've found your problem.