Here’s my take on it. Firstly there are two different scenarios to be considered:

1) You and possibly others search for a particular trip. Airline notices these searches and increases prices to everyone for that particular trip.
2) You search for a particular trip. Airline notices your search and increases prices only to you for that particular trip.

I think (1) is plausible. In other words, airlines could adjust prices based on searches rather than simply on reservations. As a crude example, if it looks like Man Utd might get into the Champions League final, lots of fans might search for those flights but not want to book (and therefore affect inventory) until the semi-final is over. The airline might pick up on the searches and pre-emptively change the inventory and/or pricing even before anyone has actually booked.*

I think (2), which is what this thread is probably about, is a myth. Firstly, I think it would be unlawful at least in the UK. In my view there’s an implicit representation that a fare that is actually available at a particular time on, say ba.com UK site, should be available to all users of that site while it remains available, i.e. ba.com UK should not offer different prices to different users for the exact same inventory. Secondly, as others have said I don’t see much benefit in doing so. Anyone can just clear cookies or look elsewhere. Internet users are getting much more savvy about these types of things.

As for redpalmetto’s example, I think that’s just a matter of inventory coming out when you and possibly others are doing dummy bookings. It’s not, in my view, intentional.

*Obviously for something like the Champions League final the airlines will be aware in advance anyway, but there will be some situations where they aren’t aware and could devise a system which hikes prices when there is a surge in interest alone, rather than a surge in actual bookings.