As is evident from the fact that there is a half-burning tree, the landscape that Peredur is in is somewhat magical. That is pretty much what the tree symbolises: it's a sign that he's left his own world and "isn't in Kansas any more."
As explained in Will Parker's Mabinogion.info:
This extraordinary vision strongly hints at the allegorical, but its significance is never explored or disclosed. As with so much in the dreamscape of Romance, it is presented but not explained. Rather than being a token or a parable, the burning tree is a monstrum – a sign or a prodigy – that suggests that the protective boundary between the internal and the external worlds is in temporary abeyance. Such wonders serve to emphasise the apprehension that the final phase or end-game of the magical plot is close at hand.
I can't remember where I read this, but I've also heard a more subtle interpretation of the burning tree: because the tree is both burning (supernatural) and living (the "real world"), it symbolizes the connection between the two worlds. This fits with the other image that accompanies the burning tree:
And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep would cross over and become white; and when one of the black sheep bleated, one of the white sheep would cross over and become black.
(Source: Peredur the Son of Evrawc)
Like the sheep, Peradur easily moves between the real and spiritual world (i.e. changes from black to white).