I'm not a nurse, but I am a manager (in the marketing department of a business credit company).
I would never, ever be expected to pick up the slack for the people who work for me. I get paid more per hour. It's a money loser for the company were I to do such a thing.
So tell your boss that. To throw out fake numbers just so you can see the math, let's say you make $100/hour and your staff make $10 or $25/hour. For simplicity's sake, we'll say you only have 2 people under you.
When you work 10 hours, it's $1000.
When each of your 2 staffers works 5 hours apiece, it's $50 + $125 = $175. If you're doing their jobs, you're costing the hospital (I assume it's a hospital setting you're in) an additional $825 and you're not even getting your own work done. Even if you rushed to get your 10 hours' worth of work done in 5 (I mean your own work and not coverage), it's another $500.
By covering for your staff, it's costing the hospital $1325 ($825 + your extra $500 in pay because you need to work OT to get your own stuff done).
They're losing money on this proposition. Claiming there's no $$ in the budget to hire is nonsense. They have the $$ to pay the discrepancy; they're just too lazy or disorganized to put out an ad and search for someone.