Two potential problems arise when washing dog bedding or a soft dog basket. First, if the bedding is full of hairs you need to remove as much as possible before washing or you could end up with a blocked pump filter full of hairs. Try shaking it outside or brushing them out with something appropriate. Wet rubber gloves are reported to be effective at scraping them off with your hand.
The second problem is balancing the load. If your pet just uses old blankets try to add some more old blankets in with them to balance out the load. If not, either the washing machine will refuse to spin because it can’t balance the load, or it may spin violently out of balance if it’s an old model and cause damage
If the dog bedding is one of those soft manufactured dog beds have a look to see if it has a wash label, which would confirm it is safe to wash and give appropriate temperature and spin advice. It could even say do not spin, in which case it would have to dry out on its own. These manufactured baskets can be awkward to wash depending on size.
It may just be too awkward to wash unless you have a washing machine with a larger than average drum size. If it’s a small one it may be perfectly ok in the washing machine. Keep a close eye on it during the wash process, especially at the spin stages, and if the label says do not spin make sure the washer doesn’t spin.
If the washing machine refuses to spin the bedding it probably just can’t balance it. Either accept that and let it drip dry, or put extra old towels or sheets in with it to try and balance it out. If you have an old washing machine it may not have out of balance protection in which case it may allow a spin but if not balanced it might be very violent and cause a lot of damage to the washing machine or even surrounding cupboards or worktops.
Related: For more details on how out of balance spin protection works and why some loads won’t spin read Why wont washing machine spin just one item or very small loads?