I don't count as "mistakes" aspects of the plays that appealed to Elizabethan audiences, but that are less suited to our modern tastes. Rather, I'm referring to issues that arise in theatrical productions cross-culturally and across centuries.
Henry IV, parts 1 and 2 is prone to one such mistake: dead space onstage. People onstage waiting, or staring into space, sap energy from the scenes. Vast swaths of emptiness where the play calls for hub-bub can have the same effect.
Unfortunately, Shakespeare spring-loaded the Henry plays with this trap. The plays include many pub and market scenes, where many people must be onstage, but only one or two (typically Falstaff and Prince Hal) are talking.
The plays also include battle scenes where, implausibly, only two people are onstage. Worse, the plays grind to a halt for inopportune monologues by Falstaff - e.g., his monologue on honor, just before a major battle; his monologue on sack, redundant and slowing of the already slow pace of Henry IV part 2.
These scenes simply don't work as commonly staged. A pub containing people standing around, watching two people talk, doesn't come across as a real pub. Where are the ribald conversations? The games of chance? The flirting? Nor does a battle scene with merely two people in it work. Where's the noise and smoke of the battle? The movement of fighters and animals across the battlefield? The chaos of war?
As for Falstaff's soliloquies, the most promising way to minimize their plot-dragging tendencies is to set them in context - in the swirl of battle preparations, for example - rather than the clear the stage and ask poor Falstaff alone to bear the weight of the entire audience's expectation.
I appreciate the exigencies of cost and the pragmatics of staging a scene so that everyone in the audience can see it. Nonetheless, there's no point in having an enormous cast (as one must for the Henry IV plays) and keeping them backstage when they could be put to work onstage. Nor is there any point in staging a scene that is visible to all, but compelling to none.
The pub and market scenes need real activity - waiters buzzing back and forth, patrons up to their own tricks, pub owners disciplining staff, pickpockets. The battle scenes need real action, whether offstage in sound or onstage with other fighting or troop movement. And Falstaff, sociable creature, needs people around him.
Otherwise, one ends up uttering of the bulky Henry IV plays what Prince Hal cries upon mistaking Falstaff for dead: "could not all this flesh/Keep in a little life?"
(Image of Roger Allam playing Falstaff in the Shakespeare's Globe Theatre's 2010 production of Henry IV, part 1 from The Telegraph)