Short answer: Yes, see The 60-Day Challenge.
It will get you to a base level of strength, and help burn fat – assuming you don't overcompensate with the amount of food you eat.
However, I would look into body weight exercises to help the rest of your body. For example:
- Back extensions
- Glute-ham raises
- Reverse rows
When you get strong enough you can combine the pullups and dips into a muscle up. (which can also be performed with a straight bar).
Body weight exercises are compound exercises, which means they exercise a group of related muscles together. That helps build strength. Having started incorporating more body weight exercises I'm gaining even more respect for them. However, you do want to keep major muscle groups involved:
- Core (abs and back)
- Legs (both front and back: pistols)
- Back (pullups and muscleups, reverse rows, back extensions)
- Front (pushups)
Your arms will get more than enough attention from all of these exercises, so you really don't have to do anything special for them.