* It encourages people to pick the best, most expensive lawyer in town for every case no matter the expense, because your opponent will be picking up the tab.
* It discourages people from using the courts to resolve their disputes, which means people without resources tend not to get any justice. If, say, your landlord steals your $1500 security deposit for no good reason, are you really willing to risk $5000 in lawyer's fees over $1500? Of course not, you'll let it slide and some little injustice will remain unresolved.
* It makes the courts a weapon of last resort rather than the standard dispute-resolution mechanism. You used to see people dueling in the streets to resolve their problems.
The disadvantage of course is that people can raise disputes with you, and you have to answer them much the same way you can't ignore filing your taxes.