If you have more units that depend on a strategic resource than you have of that resource, you will suffer a significant combat penalty, see the "Strategic Resource Penalty" line item below:
In this case, you will see a negative number on the top status bar by the strategic resource you are lacking. It does not matter how your strategic resources get into the negatives - you can have a trade deal expire, lose territory, or have an improvement pillaged, the effect is the same.
50% lost strength is a severe penalty - it makes these units worse than contemporary units that don't require resources (ie, it makes a Swordsman worse than a Spearman) and can even make the unit weaker than units of previous eras.
The quantity you are over doesn't seem to matter, either. You can be -1 or -5 with the same 50% penalty. I tested this just now on a hotseat game between two players that were running different difficulty levels. The difficulty level did not seem to impact it either. Further, I tried two different units, a Swordsman and a Roman Legion and both had the same penalty applied.
Another side effect of going into the negatives is that any unit you were building that depends on that resource will immediately cancel. When I reconnected the strategic resource, I was able to resume production where it left off.
I don't believe patch or DLC matters, but for the record this was in a game with all the current expansions/DLC (Civ5 Gold, Gods and Kings, Brave New World) available.