They don't share a border, and both are non-Arab/non-Sunni Muslim countries in a largely Arab/Sunni region. I would have thought that they would be allies, but instead they're like cats and dogs.
The first thing to keep in mind is that the Iranians have a consistent foreign policy of portraying themselves as advocates for all Muslims, not just the Shiites. They do favor Shiites factions in places like Iraq and Syria, but they also have a close relationship with Hamas (a Sunni group), and much better relations with Armenia (Christian) than Azerbaijan (fellow Shiites). There is no doubt that they sympathize more with fellow Shiites, but Persian civilization has been around long enough that they understand the need to put geopolitics before sentiment.
All Arab and Muslim states have to express solidarity with Palestine, even if they actually have close dealings with Israel. The Jordanian king slams the Israelis regularly, as do the Egyptians, the Lebanese Christians, and more. The Iranians use more charged rhetoric than most. I suspect that this is for several reasons:
As you said, Iran and Israel do not share a border, which also means that Iran is far from the Palestinians. There is a general rule that the further away you are from a problem, the more you can say without actually doing anything. Making clear their stance in favor of the Palestinians and against Israel is an easy way to win support from the rest of the Islamic world and overcome the fact that they are neither Sunni nor Arab, without actually doing anything substantial such as absorbing large numbers of Palestinian refugees.
The Iranians have geopolitical ambitions, deeply rooted in history, to gain influence from Central Asia to the Levant. Most states in this region are weak and fractious, and therefore easy to manipulate. Israel is one of the few exceptions. It is small, but strong, and difficult to penetrate. With countries like Iraq, Lebanon, and even Saudi Arabia, there are various factions (particularly Shiites, but also Christians and other non-Sunni, perhaps non-Arab minorities) to patronize. With the Israelis it's more difficult. Pressure can be applied via Hamas and Hezbollah, and further support can be gained from various quarters through strident anti-israeli rhetoric, but ultimately the Iranians are unable to be as deeply involved in Israeli affairs as they are in other countries. Israel is a geopolitical problem for them, and a more difficult one than most others they've encountered.
The biggest threat to Iranian ambitions in the region is, without a doubt, the United States. The Gulf Arabs have weak militaries, and Iraq, which had fought a brutal eight-year war with Iran in the 1980's, is in chaos, some of it of Iranian direction. The US is the only power in the region that can block Iran militarily, though US willingness to do so is in some doubt. By tying Israel to the "Great Satan"--not a difficult task, considering that the Americans maintain a close relationship with the Israelis for their own geopolitical purposes--Iran can hope to undermine support for the governments that host the US military. The most prominent example is probably Bahrain, headquarters of the US Fifth Fleet, which had violent Shiite-led protests against the Sunni monarchy during the Arab Spring.
Follow-up question: was Iran friendly towards Israel prior to 1979?