Is it safe to post the entire syslog from my PI to the web? Could someone use the data to hack into my pi? Could it make my network more vulnerable to attack?
Syslog contains data in cleartext. So whatever a process deemed fit to be logged could be read by anyone. It could however be argumented that it would be very unreasonable for any service to log confidential data such as credentials but do you really want to put your trust on that?
Even in the clear there could be data that you might not want to distribute widely, e.g. connection data of your login requests (stating your IP and time). While that wouldn't make your Pi vulnerable per se, it would still tell something about you.
I wouldn't recommend it. Considering that syslog is used to track sensitive system information, there's no sense handing out such information to the general public. If nothing else, you should redact sensitive information that is not relevant to the problem you're working with. As @Ghanima notes, it's really up to individual programs using syslog to decide how much info to dump, and this may vary depending on settings (e.g. debug or verbosity settings).
A lot of sensitive information has been revealed on pastebin and similar sites by people troubleshooting problems.