We study burst erasure correcting streaming codes for three-node relay networks, where there is a source-relay link and a relay-destination link. These codes guarantee that all message packets are recovered within a delay of $\tau$ time slots, given that a single burst erasure of length at most $b$ packets occurs in both links. Leveraging previously known techniques in the streaming code literature, we first provide a simple upper bound on the rate of burst erasure correcting streaming codes for three-node relay networks. Our main result is a coding scheme that achieves rates arbitrarily close to the rate upper bound as message size increases.