Hygiene-wise, need to clean grill/broiling tray after each use?
For the grate in an outdoor grill, I think you can feel safe not cleaning it, but only under the following condition: Before each use, you heat the grill to a temperature in excess of 250 degrees (F) for at least 15 minutes. Gotta heat it up to make sure you kill everything that might have got onto it since you last used it.

Since you really ought to heat the grate up pretty seriously before trying to cook on it anyway, this is almost a no-effort thing.

In terms of extending the life of your grate, it's good to give it a good burn-off AFTER cooking if you're not going to clean it outright. Burning off will make sure there aren't any wet or corrosive things still left on the grate.

As to the broiler pan, I would never let that one go unwashed. Unlike a grill grate, you don't preheat it, so you don't have a chance to kill the beasties that might have grown up in whatever you left on it from last time. If you cooked a steak yesterday and didn't wash the pan, the grease and juices have had 24 hours to attract and breed bacteria and whatever by the time you use it today. Then you pull it out, plop another steak right on your bacteria colony, and broil. What's exposed to the heat directly may get hot enough to kill germs, but what's directly under your steak almost definitely will NOT.

Will you get sick if you don't clean your stuff? Maybe not--it's your life. But please, please, please, don't be so casual about cleaning if you're cooking for ANYBODY else. And really, cleaning a broiler pan isn't that hard. You can man up and just do it.