Rabbi Frand once dealt with this problem by speaking of the case of an answering machine or a FAX. While leaving a message on an answering machine in Eretz Yisroel would not be a problem as long as the recipient did not actually use it, you could cause a problem if the person at the other end would actually be mechalel shabbos as a result of your leaving the message. He said that many people would turn off the answering machine so as not to be faced with the problem. A similar answer would seem to apply with the web site.
The moderators of Mail Jewish will often not handle messages even though it is not Shabbos where they are in order to avoid a possible mar'is ayin situation or to avoid leading to someone being mechalel shabbos by sending a message on shabbos to recipient who is not in the shabbos time zone.
Consider someone for whom Shabbos has already ended with a server in an area where shabbos is still extant. Even though the posting is automatic and not done on shabbos, the time stamp could bring about a situation of mar'is ayin.
Another point to consider is that we really do not know where any web site is actually hosted or the path a message actually traverses to get from the originator to the recipient. This is similar to leaving a snail mail message in the mail box on Friday. This is allowed since you are not makpid on its being picked up and carried on Shabbos. Another analogy would be how you handle snail mail delivery on Shabbos.
time-zonesince as far as I can tell Israel is just an example here and the same question would apply to e.g. China and Belgium as to Israel and the States. By all means revert me if I'm mistaken.