Increase Provisioned IOPS in RDS leads to downtime?

There will be no downtime according to Amazon. Reference below.

IOPS: The new Provisioned IOPS (I/O operations per second) value for the RDS instance. Changing this setting does not result in an outage and the change is applied during the next maintenance window unless the ApplyImmediately parameter is set to true for this request.

Default: Uses existing setting

Constraints: Value supplied must be at least 10% greater than the current value. Values that are not at least 10% greater than the existing value are rounded up so that they are 10% greater than the current value. If you are migrating from Provisioned IOPS to standard storage, set this value to 0. The DB instance will require a reboot for the change in storage type to take effect.

ModifyDBInstance

So, after @John Hanley's answer (and testing on a test instance) we went ahead with the change.

The Operation took around 1 hour and 6 minutes. Here are the relevant charts from the period.

We faced no downtime during this period. And even though the total IOPS (~1600) was higher than our old limit (1000) we didn't face any degradation in performance.

Thought this will be helpful for people in the same situation.