"Routine" storage, as you clarified, would assume that you're running the machine every few weeks. In that case, the only reason to cover it at all would be because rust-prone metal is exposed on the top exterior.
Modern mowers have powder-coat paint on any exposed steel surfaces except for the blade(s), and all other metal either won't rust (e.g., aluminum or zinc) or is shielded by plastic covers. If you have chipping paint or failed galvanizing you should repaint to protect the steel.
Any covered or steel internals of the engine will be protected by running it every few weeks, which will heat it up, dry them out, and coat them in the engine oil.
So the most rust-prone piece of a lawn mower is usually the blade(s), and it's unclear whether covering it will protect those. Over a period of days large swings in temperature and humidity can produce condensing conditions even in the absence of rain. Covering a machine stored in unconditioned air with a tight tarp is just as likely to exacerbate condensation as to mitigate it. (For example, watch your lawn: You'll notice that the sections that get hit first by the sun dry out the best, while sections left in shade can retain dew much longer.)
Most of us just accept that the blade will rust. So we sharpen and balance it as necessary. If you really want to stave off rust you could spray or wipe the exposed steel of the blade with a rust preventive liquid after each use. But unless you store your mower upside down covering it will not help protect the blade.