Should I use a case on my MacBook Air to prevent hinge damage?

Apple does not recommend cases for any of its laptops.

The MacBook Air is a fanless design that relies partly on its aluminum skin to keep its operation temperatures under control. A case will inhibit that cooling.

Best not to tempt fate.

Keep your Mac laptop within acceptable operating temperatures - Apple Support

I would recommend a soft sleeve be used when you need to transport the computer.

It's not necessarily just the hinges that can be damaged...these types of cases use clips to hold them onto the MacBook. Today's MacBooks are quite fragile and the clips can cause stress on area of the case (especially the lid and it's delicate LCD panel and front glass assembly). There are miniscule tolerances between components of the screen assembly that can be damaged by these clips that are meant to hold the case onto the lid and bottom case. Simply removing the case after it's been installed could cause damage trying to unclip the clips. So the case that was meant to protect your MacBook may actually damage it when it comes time to remove it.

The heat dissipation is another aspect of it that was brought up by our friend D.I. Johnson.

Bottom line is that it's not recommended for various reasons noted above.

I agree with D.I. Johnson about using a simple padded bag or sleeve for protection.