This really all depends on what kind of game you're making. Not only the genre, but how you're presenting it: 2D top-down, 2D side-scroller, 2D isometric, 3D first-person, 3D third-person shoulder, 3D third-person isometric/freecam...
Basically, from an artistic standpoint, rather than an actual game-mechanic standpoint, there are several things that you can do. Things like palette-changes, lighting-changes, starker/longer shadows, visual-noise, tightening the camera (sense of claustrophobia), shaky-cam...
...switch the character's running speed, slightly, change idle/movement animations to depict that the characters, themselves are nervous.
Generate spiders or bugs, or whatever is creepy and fits within the theme. Make them tiny, non-interactive, but make tiny visual cues of some sort. Not a BOO factor, but a creepy factor.
I can't tell you how to do it in YOUR game, because there are too many variables we don't have, both in theme and in presentation, but you should watch some old horror movies or old psychological-thrillers, et cetera. Avoid newer movies, because, surprise, after the mid-'90s, most have been "slasher-horror" with pop-up scares. Pay close attention to what mechanics worked for film-making in the old days.
Sound was a big deal! Not just music, but sounds and sound-themes (nature or sewers or empty streets, etc). So were pacing and lighting and camera angles (I suggest looking at Evil Dead 1 -- the effects are dated, but some of the shots are still mind-blowing).
Unless you're doing a lot of cutscenes or controlling the character movement like in a Point'n'Click setting, all of these things aren't always available in a 1:1 translation to your game's format, but whether they come through as special effects, or as cutscenes, or artistic direction for level design, or pacing in terms of ramping up dangers or ramping up numbers or difficulties of enemies... each or all of these things can improve the sense of tension.
If you're going to put insta-kill traps out, then put some warnings out, maybe a pre-sprung trap with a corpse in it.
So yes, music, sound-effects, soundscapes (sometimes the most-effective "tense" scenes are the ones where the tense music goes away, and you're left with a creepy soundscape), but also character-animation, colour-palette changes (just look at the average Super Mario World level versus colours inside of a haunted house), camera-angles, decreases in visibility (fog of war, or zoomed-in camera), increases in shadow-intensity and shadow-length, artistic creepiness (spiders/bugs/intestine-covered walls/alien growth/etc), visual cues to increased danger (pre-sprung traps/previous adventurers/etc.)... the more of these you put together (especially when you mix pacing with visual with audio), the more tension people are going to feel.
The question, of course, is how to pull it off in your particular game. And really, it's just an art which needs to be practiced and tested and revisited, because there is no one correct mix.