Should I trim the brown tips off?

Yes, with clean scissors. Cut inside the brown, leaving a sliver of it behind, so you never wound the green tissue. The leaf keeps its shape and the damage stops looking ragged.

Do brown tips spread to other leaves?

The brown patch itself does not spread, but the cause does. Dry air, salt build up, and uneven watering keep marking new leaves until you fix the underlying habit.

Brown Crispy Tips

Humidity

What is going on

Dry brown edges and tips creeping inward. Indoor air is the usual suspect. The full guide walks through the checks in order, fastest first, so you can find the cause before trying a fix.

The fix, in order

Start with the check that costs nothing: light, water, and where the plant actually sits. Most cases of brown crispy tips trace back to one of those three, and the guide for your specific plant will narrow it further.

The step by step fix

Check the water before the air. Most crispy tips start with watering that swings between drought and flood. Feel deep in the pot, and if the plant has been drying out hard between drinks, move to a steadier rhythm first.
Look at where the plant sits. Radiators, heating vents, and a fierce sunny sill all bake leaf edges. Move the plant out of the direct line of hot or dry air.
Flush the salts. If you see a white crust on the soil surface or the pot rim, minerals have built up. Run plain water slowly through the pot, about four times the volume of the pot itself, and let it drain away completely.
Raise the humidity properly. Group plants together, stand the pot on a tray of pebbles and water, or run a humidifier nearby. Steady humidity fixes tips; a daily misting mostly does not.
Trim for looks. Cut the brown away with clean scissors, following the shape of the leaf and leaving a sliver of brown behind. Cutting into green tissue just gives the leaf a fresh wound to dry back from.
Watch the new leaves. Old tips will not heal, so judge the fix by fresh growth arriving clean over the next month.