The bigest animal shelter of the canton is in Oberbottigen, a good 30km from Thun... They don't mention that they search volunteers for walking the dogs or helping in the shelter, but you could ask them by mail (info@bernertierschutz.ch) if there is need for some helpers.

The canton's animal protection club also maintains a list of volunteers who are ready to look after the pets of those who have to can't look after them short-term (because of vacation, hospital stays etc.) for no or little compensation. The registration form is on the website.