A Processing app running on Android is actually an Activity. When the activity loses focus it almost always stops. Even if you use the home button, it still doesn't keep running in the background. The only way that I can see is to write a full-fledged program in Eclipse.

I have a similar issue, would really appreciate some tips on where to go from here.

I don't pretend to be an expert, but I believe it would be possible to run your Processing Activity within a "real" Java program... Or you could go through and rewrite Processing so that it is not an Activity... Or you could write a Java program that imports processing.core... Just some thoughts, maybe not all that realistic.

I'm also trying to do this. I want to figure out how to run an application in the background that continuously logs sensor data. Does anyone know if this is possible without having to reprogram the entire application in eclipse? 

I am no expert, and I understand this thread is old, but for those who still would like an answer or something close, I found this:  http://developer.android.com/guide/components/services.html

calsign is completely right.