Where to use on and upon?

Although this definition suggests that "upon" is simply a one-to-one replacement for "on," there are some constructions that don't work with both. "Take on" can mean "adopt," "take on [-to myself]," so the second example would be either "I took on his emotional burdens" or "I took his emotional burdens upon myself."

In the third example, in which "on" is synonymous with "about" or "with regard to," "upon" is idiomatically inappropriate. (If you said, "Upon the topic of __," it could be appropriate, but as with just about any valid use, somewhat archaic, as the definition above notes.)

In the first example, you could change "on" to "upon" with no change to meaning, but definitely an archaic usage.