Strong AoEs or attacks that do a wave or some sort wipe out illusions pretty quickly. Keeper of Light is a prime example. Phantom Lancer is also a mid-late game carry, so the best way to deal with him early on is to gank him and shut him down early.
If I recall, Black King Bar will protect against naga's siren. And paying attention on where the enemy team is, since they'll usually be hiding in the jungle, letting Siren go first to initiate. Being prepared does wonders.
Most of the conventional answers (AoE) aside, Shadow Demon can turn PL illusions against him since the passive ult can proc on Shadow Demon illusions. You can also create illusions from the illusions you already have under you control. Although it might be a bit beyond a new player to pull this off since it involves macro and the ability to know how to use the illusions effectively, but it's much better than pure AoE IMO because Phantom Lancer can just create a new army very quickly.