I think initiative checks are vastly underused. I think that the DM should
Roll Initiative Early & Roll Initiative Often
If you're the DM and following the advice below, be absolutely clear with your players that initiative checks are commonplace before the campaign begins!
Pigeonholing initiative to actual combat instead of potential conflict needlessly removes a valuable DM tool for adjudicating the order of actions. Initiative should be rolled whenever the order in which participants take their actions is important. The DM can liberally define battle as conflict then call for initiative checks for everyday, non-adventuring occurrences that involve any kind of conflict.
The DM can call for initiative checks when...
- one or more creatures whose intentions are unknown approach another creature (e.g. "The city watch approaches your position; make initiative checks").
- one or more unknown creatures merely approach another creature (e.g. "The froghemoth wades through the muck toward you; make initiative checks").
- one or more creature enter another creature's line of sight (e.g. "A farmer leading a cart on which is a cage containing a single penguin rounds the corner; make initiative checks").
- two or more creatures have the opportunity to cause the same event (e.g. "Everyone has a bomb, but there's only one penguin; make initiative checks").
- two or more creatures have the opportunity to prevent to the same event (e.g. "The timer says it's 12 second before the penguin explodes; make initiative checks").
- an unanticipated event occurs (e.g. "The penguin explodes; make initiative checks").
This means the DM can ask for initiative checks when an urchin (street not sea) in town approaches the PCs, begging for money, and the PCs can whip out their weapons and murder the poor orphan before he's even said, "Please, sir." Or they could, y'know, delay and see what he wants. Or they could ready an action in case he's a pickpocket... or a disguised assassin. The world is a dangerous place. PCs fight shapechangers, teleporting wizards, ghosts, and other unspeakable evils--they're allowed to be always on guard. Potential conflict is everywhere.
Of course, a DM would be foolish, time-consuming, and metagamy to call for initiative checks all the time, but the PCs should be aware that drawing a weapon isn't any more of an indication that actual combat's going to take place than a snarky comment from a vagrant or a seductive glance from a hottie across the room, but PCs should also be aware that all of these can be signs of potential conflict which often requires the DM to impose some kind of order on the world. One of the tools the DM has for imposing that order is initiative checks.
The DM should ask questions like, "Are you wary of the creature?" or, "Do you think something's going to happen?" and if the answer's Yes then call for initiative checks, but insist that combat needn't ensue. Make the player, not you, responsible for the conflict. If both the PC and the creature are aware of each other, and the potential for conflict exists, the battle has started, just no one's fighting... yet.
Initiative isn't an excuse but a tool.
You asked, "If my PCs are in a tense situation, and an NPC suddenly draws a weapon, should I ask my players to roll for initiative?" To which I say No. Instead you should've had the parties involved make initiative checks when the tense situation began, before weapons came out... and then even if they never come out. The battle started when things got tense and both sides are aware of each other, so initiative should've been determined then. You probably should keep numbers secret until hostile actions are taken, though.