No, but failing to record can cause serious problems.
A lady bird deed takes effect once it is properly signed and delivered to the grantee, the person receiving the future interest. Delivery can be shown in several ways, but the most common way to create a presumption of delivery is to record the deed in the county official records.
Recording also matters under Florida's recording law. An unrecorded deed is not effective against later buyers or creditors who have no notice of it. Recording puts anyone with a claim against the property or the owner's estate on notice that the future interest has been transferred, unless the owner signs a new deed later.
An unrecorded lady bird deed found after the owner's death can also be hard to prove, which can send the property into probate. QuickDeeds offers eRecording for an additional fee.