I would not recommend cardboard only. If it’s a temporary placement, you can use it but line the entire inside with a good quality, reflective tape. This is aluminum foil duct tape made for HVAC applications. This will shield any heat from getting to the flammable cardboard as well as give it some stability.

Nothing serious will happen. It’s just going to cut down the efficiency of your system – something you never want to do as it will place higher resistance on your blower. This could shorten its life. Also over time, less air can move through, and your house won’t be conditioned as quickly. It’s best to make sure your plenum is in good shape (no dents, build-ups of any kind, etc.) so your air can flow as it should.