If your goal is increasing strength, must you always train to the point where you feel the burn later?
Probably not. Training for powerlifting (completely strength-oriented) involves avoiding failure, keeping the number of reps low, and resting a lot between sets, none of which are really conducive to "burn" during a workout or soreness afterwards.
Pavel Tsatsouline gave this answer to a similar question during an interview:
One thing is for sure, you can’t use soreness as an indicator of progress. The only meaningful gauge is the increase in your strength or your muscle mass. Soreness doesn’t mean a thing.
And on the likelihood of soreness from powerlifting-style training:
If they go to low rep, heavy, non-exhaustive training — three sets of three or five sets of five — they would not get sore.
To increase strength, you must effectively break down your muscle tissue so that, when it heals, it comes back stronger than before. This means that, to some extent, you must "feel the burn".
Additionally, after the workout, you must give yourself proper nutrition in the form of protein, carbohydrates, and fat, as well as give yourself plenty of rest.
Over time, you'll feel less sore after the workouts as your body adapts and becomes stronger. With that said, you should never exercise to the point where you are in physical pain. Soreness and pain are different animals, minimal soreness can be good both mentally and physically. If you are feeling pain, that means that you're doing something wrong, and you should make adjustments to prevent the pain from recurring.
I used to believe you had to "feel the burn" for a long time, until I heard an expert (sorry but I forgot who it was), explaining that the "burn" is actually the body's signal for us to stop. It's like the safe limit, the point where lactic acid begins to build up a bit too much, and the time where you need to take a break before going again.
Natural pains after "working out", are pretty fine, though.