On 2 November 2015 near the end of the Seahawks-Cowboys game, the Cowboys were down by 1 but had possession of the ball. They had no timeouts and next to no time left on the clock. Hurrying to get a play off, they committed an illegal shift by failing to get set before snapping the ball. The result of the play was a sack, but the sack was negated because in the last minute of a half an illegal shift is treated as a false start. In other words, no free play is given for the defense. Why is this? Sure, it aids the offense in a potentially viewer-friendly, comeback situation, but it's not hard to think of a jillion other rules that would similarly aid the offense.
It's possible Pereira was referring to the section referring to illegally conserving time:
SECTION 7 - ACTIONS TO CONSERVE TIME
ARTICLE 1. ILLEGAL ACTS
A team is not permitted to conserve time inside of one minute of either half by committing any of these acts:
a foul by either team that prevents the snap (i.e., false start, encroachment, etc.)
an illegal forward pass thrown from beyond the line of scrimmage
throwing a backward pass out of bounds
spiking or throwing the ball in the field of play after a down has ended, except after a touchdown
any other intentional foul that causes the clock to stop.
Penalty: For Illegally Conserving Time: Loss of five yards unless a larger distance penalty is applicable.
When actions referred to above are committed by the offensive team while time is in, officials will run 10 seconds off the game clock before permitting the ball to be put in play on the ready-for-play signal. The game clock will start on the ready-for-play signal. If the offensive team has timeouts remaining, it will have the option of using a timeout in lieu of a 10-second runoff, in which case the game clock will start on the snap after the timeout. The defense always has the option to decline the 10-second runoff and have the yardage penalty enforced, but if the yardage penalty is declined, the 10-second runoff is also declined.
These aren't explicitly false starts, though, and not all are even dead ball fouls.
There's nothing in the actual NFL Rulebook that refers to converting an illegal shift into a false start. If it's true, and I have no reason to think Pereira would be wrong about something like this, it's either in the referee's supplement (which is not freely available) or an interpretation of something I'm not seeing. (I don't see, either, where it says that the offense must get set on certain plays differently in the last minute, which he also refers to in his tweets. The offense must always be set for one second - see 7-4-6 - so I don't see how the last minute is special.)
The complete false start rule is rule 7-4-2, and reads:
ARTICLE 2. FALSE START
It is a False Start if the ball has been placed ready for play, and, prior to the snap, an offensive player who has assumed a set position charges or moves in such a way as to simulate the start of a play, or if an offensive player who is in motion makes a sudden movement toward the line of scrimmage. Any quick abrupt movement by a single offensive player, or by several offensive players in unison, which simulates the start of the snap, is a false start.
There are then some specific items clarifying what is/isn't a false start for particular positions, but nothing about the last minute.
And for illegal shifts, rule 7-4-8:
ARTICLE 8. SHIFTS
The offensive team is permitted to shift and have two or more players in motion multiple times before the snap. However, after the last shift, all players must come to a complete stop and be in a set position simultaneously for at least one second.
If any eligible backfield player goes in motion (one at a time) after the last shift and comes to a complete stop, there is no requirement for a full second pause before a second player can legally go in motion.
However, if the first player has not come to a complete stop when the second player goes in motion, it is another shift and requires another simultaneous stop for at least one second by all players.
It is also an illegal shift if a player under or behind center goes in motion and fails to come to a complete stop for at least one full second prior to the snap, or before a second player goes in motion.
Note: The offensive team must present a legal formation both before and after a shift.
Penalty: For an illegal shift: Loss of five yards.
What I suspect, though, is that this was just badly worded. Rule 7-4-6 is probably what controls here, and while it doesn't have a specific penalty associated with it, it clearly would be a dead ball foul.
ARTICLE 6. COMPLETE STOP
All offensive players are required to come to a complete stop and be in a set position simultaneously for at least one second prior to the snap.
That describes more aptly what it sounds like was the problem: if the players were never set properly to begin with, it's not a shift. I'm not sure why a penalty isn't explicitly listed here, but it would be similar to a false start, so why not call it that. It's also not clear if it's a dead ball foul (like false start) or a live one, but dead ball makes sense to me.