According to the Urban Dictionary:
to toodle pip a dear friend means to say goodbye in a very friendly way.
How common is this expression? Can I use it without any fear to be misunderstood?
The expression was common among upper-class English people around 100 years ago but is now very much obsolete. If you use it now you run a severe risk of not being understood or of being considered foolish. The Urban Dictionary is a VERY BAD source of information for English learners.