I understand that serious HW firms can manufacture their own boards, but what are the disadvantages of using a development board 'in production', i.e. placing a PCIe card into a server and performing calculations on it?
The biggest one is that it might not be available tomorrow.
In some cases, the manufacturers intend that the development board may be used in low volume production and pledge to maintain production over some period of time. They may also make available schematics and gerbers that allow you to produce the boards yourself.
Producing a compatible board over a long period of time is a significant commitment- like making a product. They have to deal with components that go obsolete, document changes and so on.
Other disadvantages center around the fact it wasn't designed just for your application so it might be too big, too power hungry, lacking in features etc.
OTOH, the relatively high production can have advantages. The cost of a development board for an aerospace client of mine was less than the cost to just populate an equivalent bare board (that's before buying the parts or testing).
Speaking from experience...
Many years back my company was experimenting with a low-volume FPGA application. At the time there were no production boards that met our needs and because of time pressures it wasn't feasible to design our own. We ended up going with a demo board produced by a well known Xilinx distributor.
The biggest negative we encountered was a very high infant mortality rate. For whatever reason the distributor didn't invest enough in their manufacturing test process. Also, a typical development board probably has many components that your design doesn't need, but those components still contribute to the failure set.
To make matters worse the distributer wasn't really setup to handle rework and, to make a long story short, we ended up with a large pile of non-functional boards in our lab.
On the positive side, the boards allowed us to meet our production needs, and, the boards that worked at installation time tended to keep working.
In the final analysis, the project lived longer than we expected and in the latter phases we just got into the habit of ordering two cards for every one we needed.