You will find an area with flames shooting out the floor. You can't get past them without getting hurt, and one hit will take off half your health. What's going on here is that the flames are triggered by the robots walking on the floor tiles directly above. Ergo, take out the robots and there will be no more flames. You will have to repeat this later on, though if you have full health and you can handle the spikes up ahead, you can simply run through and take the damage if you're impatient.
Don't get careless while going down the ladder. Long drops will kill you, and once you start falling a lethal distance, you won't even able to grab the ladder to break your fall.
After the second set of flames, you will come across some spikes. It's easy to get hit here if you don't know how to handle them. Switch to the spring shoes and run towards the spikes. Don't jump until you are in contact with the spikes; they can't hurt you unless you hit them from above. You should land in the middle of the platform. Back up to the left edge of the platform to give yourself room, then run and jump over the remaining spikes.
If at any point you do get hit by the spikes, just use the temporary invincibility to get past them.