Yeah that heat exchanger and or the burners are plugged. DO NOT run it. Plugged = soot = carbon monoxide. With it being your rental I wouldn't dick with it. I would leave it to the land lords. Document document document. They(landlords) should've provided you with co detector per Megan's law, if you don't have one buy at least one with a digital read out. That exchanger and burners should be cleaned professionally. I recommend duct works, monster vac etc. I'd bet $5 after they clean the heat exchanger there's a crack. Cracked heat exchanger means new furnace. Pm me if you want my contact info for advice.
If you wanna get their attention call xcel energys gas emergency response line and tell em you smell gas. Once the guy gets there and sees that they will red tag that furnace in a heart beat.
Might consider sleeping in a tent to stay warmer tonight if you have one.
"There are no finger prints under water."
Looks like the heat exchanger is plugged up. Burner needs to be pulled and cleaned along with the heat exchanger. Landlord needs to get an HVAC there ASAP! You CAN NOT RUN IT LIKE THAT!!!!!
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Check out Goodwill and Salvation Army thrift stores. I scored a pair of electric space heaters that were like brand new, for $5.50 apiece. These things sell for like $50 each in stores. I use the hell out of them.