Is it safe to use one of those Brita pitchers to filter river and lake water while camping?

In a word, no. Brita-type filters are designed to take out waterborne contaminants such as chemicals, and to remove sediment. They aren't at all intended to "purify" water or remove biological nasties such as giardia.

See the full response and article by Doug Gantenbein on Outside Online's website here.