Perhaps indirectly.
A short sale is worked out between you, your lender(s), and the buyer(s). Therefore, everything is open to negotiation. The lender can request a contribution from you as a condition of okaying the deal--but you can say no. Of course the lender will want as much as it can get from you and the buyer. Just as the buyer wants the best deal possible and you would like to avoid being skinned alive.
The lender may request cash from you at closing or demand that you sign a note for part or all of the shortage. But that doesn't mean you have to just fork it over. I recommend having a real estate attorney or bankruptcy lawyer experienced with short sales, foreclosures, etc. help you with the negotiations.
If all parties have dug in and can't be moved, the short sale can't be concluded. This would then probably cost you some money--you'd have to keep making payments to avoid foreclosure, or you may have to cough it up when the lender forecloses. In most states, if the lender can't recover the balance owed by selling the property, it can take you to court and sue to recover the funds. And if you have the money, well, you may have to part with it--this is called a deficiency judgment.
But the lender does NOT hold all the cards. First, if you don't have the money it's pointless for anyone to incur the expense of taking you to court. And agreeing to a short sale saves the lender the trouble of foreclosing, rehabilitating the property, and carrying it on its books month after month. Convince the lender that it will get the most money from a bad situation by accepting your offer. If you are insolvent you could offer the lender a deed-in-lieu of foreclosure. You could also avoid a deficiency judgment by filing bankruptcy--which would protect certain retirement accounts and other assets. Again, a lawyer may be your best guide in this situation.