I had the same problem with this printer. With this fix, it went from failing every time to picking up every time.
There's no need to disassemble anything. Just remove the toner cartridge and the manual feed platform. Put a small amount of isopropyl alcohol on a Q-Tip and gently rub the first 2cm of the pickup roller sponge from side to side. Rotate the Q-Tip as you do, and you'll see it's picking up residue from the sponge. Use the other end of the Q-Tip to help dry away the alcohol, and to remove more residue.
If the roller sponge is quite dirty, you may want to repeat this process. When done, move on to the next 2cm further around the roller, overlapping what you've already done. You won't be able to get to the back of the roller without disassembly, but that's not something I wanted to start doing.
How did it get dirty? I imagine it's because it picks up toner when you are printing something double-sided. If you only ever did single-sided printing, the roller sponge touches only clean paper, but it might also become hardened or glazed over time. Also, I'd stay away from ethanol or rubbing alcohol, which would leave more of an oily residue.