Forests aren't particularly dangerous for the prepared. If you want to know what could possibly be dangerous at all given a lack of preparation or available technology/magic, then your best source is to look at wilderness survival handbooks for what the experts prepare against.
Googling for "wilderness survival handbook" gets some hits, though your local library is going to be full of great guides as well. Reading about the bright lights of survival techniques will reveal the dark shadows of potential sources of death. The first hit on the search was wilderness-survival.net, so I'm going to be looking at that as inspirations for some examples:
Panic. The first key to surviving is to keep your wits about you when the trip turns bad. Getting yourself lost, losing equipment, splitting up, being reckless and causing further injury—these are all ways that panic can increase your chances of other ways to die coming to visit. Making panic an actual danger is going to be hard in an RPG context, but depending on your system this might be the most powerful first line of crap hitting the fan. If your system doesn't mechanise the consequences of emotional responses though, and you're not sure you can pull it off with unstructured narrative without angering the players, skip this.
Water-borne disease. When you're in the wilderness you boil your water, purify it with filters, or sterilise it with iodine. Untreated water can kill you, and boy howdy do you need to drink a lot more than you can carry when you're trekking overland.
Freezing to death. There's a reason that every survival guide has entire sections on fire starting and building shelters for you and your fire. If you're in a temperate forest, night can be cold and you're might just die of exposure, so being capable of building a fire and keeping the wind/rain off it is important. Fortunately most fantasy-setting characters have a default level of experience simply because fire-building is as common as knowing how to drive in a modern setting. Still, adverse conditions that they'd rarely ever learn to mitigate for fires at the homestead – rain, wind, wet fuel – can be problems.
Dying of burns. Even if you know how to build a fire in less than ideal conditions, how to build a safe fire outside a fireplace is a learned skill. One good burn can kill you without modern medicine or healing magic.
Getting stuck. The problem with injury in the wilderness isn't just that you're injured and might need attention to prevent infection or death, but it's also that many injuries inhibit mobility. A broken leg that no-one knows how to splint can keep a party stuck in one place, which makes finding food, water, and help harder. A victim of a "simpler" injury or splinted broken leg might be able to walk, but pain, infection, and the simpler exhaustion of being injured can slow the group down significantly, potentially stretching an exit from the wilderness from a span of time that's within supply limits to something that would starve the group if it stayed with the injured individual.
And these are just throw-away examples from looking at the headings on that site! Doing this more systematically with an in-depth reading of a good survival handbook will get you even more and more finely-detailed ways that characters can be threatened by a forest.
And of course, once the wilderness has started to kill you, that's when many of the otherwise-non-threatening predators will take an interest…
