Understood.
Not an uncommon situation.
And I am not a fan of those desks with little spaces, often with a door, to house a computer.
Overall, you must keep in mind that computers and the components within require room for airflows (as you did mention) and thus cooling.
Generally the case specifications include some minimal space requirements on all sides. And case components have specific internal clearances.
If the case is too small and/or the furniture space too tight and closed in then heat may become a problem.
May need to become a bit more creative.
Google "DIY computer enclosures desks". Take a look at some of the images and then modify the search to be more applicable with respect to your requirements and...