Syslog streaming is independent of the router resources since it sends configured logs to an external Syslog server. Losing the syslog would mean that these logs go back to the internal buffer.
The internal buffer is allocated a piece of memory by the OS itself, but is user-configurable with values ranging from 4096 to approx 2 billion. However, setting a larger value for a buffer means that you are eating up into that precious memory which could be needed for other router tasks.
So, if you set the internal buffer value to a value that you feel would not starve others ( this could vary depending on where the router is placed in the network) and you lose the syslog server, you should do fine. The only thing you would lose are the logs since internal buffer is similar to a circular queue which means newer messages replace older ones.
NOTE : We have 4 syslog servers in our network, and I think that gives us a good amount of HA