Weak hydrogen bonds are important in DNA because they make replication process of DNA easier.
During DNA replication, an initiator protein unwinds a short length of the DNA double helix . Next, an enzyme known as helicase attaches itself to the unwound helix and begins to break the hydrogen bonds between the base pairs of the two DNA strands, thereby separating them apart to form a Y shape called a replication fork.
Helicase is able to easily break the hydrogen bonds because they are weak and thus this makes the enzyme's work much easier and replication can be carried out very rapidly.