I doubt it. While PostgreSQL offers a number of technological advantages over MySQL, MySQL has a lower barrier to entry and easier to understand concepts than PostgreSQL as well as being a major part of the LAMP stack, which is still the most common entry point of web development. Rather than viewing one database solution as the dominant, what we are likely to see is the diversification of database technologies with the likes of NoSQL solutions coming to prominence in stacks.
> MySQL has a lower barrier to entry and easier to understand concepts than PostgreSQL
This affirmation need some explanation. In my view, once SQL is understood, both are similar. PostgreSQL might even be simpler, as it has no concept of "engines".
There are some changes happening, but certainly MySQL will dominate pre-existing installs for the foreseeable future. Switching databases is not trivial for most applications.
Its a small point, but I feel that MySQL administration via the console is easier to get into (for example, having SHOW CREATE TABLE to dump table creation SQL is operationally simpler than having to exit out of the console, then run pg_dump <db> --schema-only <tablename> and then start the console back up). MySQL does a lot of simplifications in that vein that I think will prevent displacement for some time to come.