In my laundry room I have a closet with a gas hot water heater and on the wall to the outside in this closet are two vents going outside. This obviously presents a heat leak as you can feel the cold air coming into the laundry room. I would like to close this up but I am not sure if that would present an issue due to the hot water heater being gas. Is there a reason those vents are there and is there anything I can do about it?
Water – Vent to outside in gas hot water heater closet, can I seal it
hot-water
Related Solutions
I've been looking into a heat pump replacement.
You said "However, when the heat pump is in heating mode, the exiting refrigerant line is actually colder, not hotter, than when it enters. As a result, you cannot heat water during the winter."
That will depend on the location of the internal connections. The HP models I have seen with a desuperheater ( hot water generator ) place the HWG after the compressor before the reversing valve. That means the pump always runs forward, the HWG always has hot refrigerant to use and it works in summer and in winter. The reversing valve does just that - in summer it sends the leftover heat to the outside and extracts heat (cools) the inside. In winter it sends the heat to the inside and extracts heat from the outside. 
New Answer Derived From Exhausting Initial Answer & more:
It's your Water Heater, here are some things to try in determining whether or not there's actually a problem.
1 - Drain & clean the tank properly. ONLY do it when the pressure's good. Turn off the Hot & Cold Water & WH, open the T&P valve & drain the tank (see manual) to somewhere that contents can be seen. Blast on the cold water when you can start to "slightly" rock the tank to stir & break things up & periodically continue this until it's empty. Operate the hose bib handle to crush any big particles or Poke into your fully open hose bib if the hose flow drops. Then, when empty blast on the cold water & let drain & repeat until nothing comes out. Then, do just a 2-minute fill & drain & blast again to get all remnants out.
2- Check your status lights to your manual for detected problems.
3- Status Lights will also indicate a problem with the heater's Gas Control valve. But, this could also be a Gas Meter response to a run-away or leak detector, improbable since flow-rate wouldn't have changed.
4 - This is a pretty cutting edge unit you have & everything's interconnected. The fan & burner won't outrun each other & there may even be a Water Exit Regulator involved to turn the heater, sort of, into a Tankless Water Heater.
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Original / Initial Wrong Answer: If you have a Hot Water Recirculating Pump somewhere else, you're driving its thermostat batty & it's only now showing signs of needing attention. It would be stealing pressure until it's reading hot water again. You may have noticed the pressure returns after 10 to 30-minutes, when the Water Heater has recovered enough or recovered fully.
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Best Answer
Your gas water heater needs air to burn the gas. Without the vent, there's a possibility of air being sucked in through the flue instead, which could fill your house with carbon monoxide. So yes, you need the vent.
The exterior vent should have a flap/door on it that closes when there is no need for additional combustion air. If yours doesn't, then get it replaced with one that does.