We know that, historically, the Sri Lankan LTTE (Liberation Tigers of Tamil Eelam) had an air force.
Given that Hezbollah has more power than the Lebanese government and the Lebanese government seems to rely a lot on Hezbollah, why doesn't Hezbollah have an air force yet?
An air force is totally unsuitable for a guerrilla-type force fighting a dominant adversary.
Which in the region is the IAF, which, outnumbered, made short shrift of even the established Arab air forces in 1967 and 1973 (in contrast, the Sri Lankan air force was not remotely in the same ballpark).
Per Mao: a guerrilla force's main advantage is to be like a fish swimming in the sea of the population. You just can't do that with aircraft (and runways as per the other answer).
Modern military aircraft are sometimes measured in the cost per hour to fly. Those costs are usually in tens of thousands per hour.
Military pilots need constant training and it's easy to guess that "accidents might happen" if Hezbollah Air started flying around. Just like Israel tries to keep a lid on Hezbollah supplies by sometimes striking in Syria.
Ditto the logistical "tail" of an aircraft. That's easy to disrupt as well.
i.e. you just can't acquire an aircraft, store it somewhere and hope to achieve something with it when hostilities start. It is an ongoing, costly, and visible process.
Iran flies some really, really, old stuff. They will likely take the first pick of anything newer they can lay their hands on (like some Russian jets they are due to get) rather than supplying Hezbollah.
Lebanese government seems to rely a lot on Hezbollah
Yes, one shouldn't underestimate the support of the general population for Hezbollah in "countering Israeli aggression" (well, rather than courting Israeli aggression). But in general, Hezbollah is highly Shiite-aligned, representing about 30% of the population's interests in a country with a history of civil wars.
Outside of its Shiite constituency, the government's relations with regards to Hezbollah are due to weakness, rather than choice (remember how closely they are aligned with Syria, a country which also occupied Lebanon for years).
Lebanon as a whole is unlikely to be too interested to support Hezbollah Air.
Air forces against Israel were more or less nullified during Operation Focus, part of the 1967 war.
At 07:45 on 5 June 1967, the Israeli Air Force (IAF) under Maj. Gen. Mordechai Hod launched a massive airstrike that destroyed the majority of the Egyptian Air Force on the ground. Following Syrian and Jordanian attacks in retaliation, the Israeli Air Force proceeded to bomb air bases in those countries. By noon, the Egyptian, Jordanian and Syrian Air Forces, totaling about 450 aircraft, were destroyed. It was also very successful in disabling 18 airfields in Egypt, hindering Egyptian air operations for the duration of the war, and remains one of the most successful air attack campaigns in military history.
Air forces are highly vulnerable on the ground. Even if you manage to get them in the air, you need hangars, maintenance, air fields, etc. All of these things can be easily destroyed by long-range weapons or Israeli fighters. This is likely why Iran and its proxies rely so heavily on rockets and guided missiles, which require far less targetable infrastructure.
The simple answer is that Israel does not allow them to have an airforce or otherwise to grow too strong to pose a serious threat to Israel. The military action (like dismantling of the Iranian "ring of fire") is only a small part of it - from the very birth of Israel much diplomatic effort has been diverted into upholding the balance of power, via preventing Israeli adversaries from acquiring weapons that could reduce Israel's comparative military superiority. This notably applied to weapon purchases by Egypt, Syria, Lebanon, and more recently Iran - both in the times of Cold War (when Soviet Union actively supplied weapons to Israel's adversaries) and in more recent times - see, e.g., Russia completes delivery of S-300 system to Syria, Russia lifts ban on S-300 missile system delivery to Iran, Israel’s strikes on Iran reportedly hit air defense systems protecting energy sites.
Iranian arm shipments to Hezbollah during the Syrian civil war
As an example directly related to Hezbollah: during the Syrian civil war Israel has actively opposed Hezbollah from acquiring more sophisticated Iranian weaponry, carrying systematic airstrikes against the weapon shipments, see Israel's role in the Syrian civil war:
Israel's military activity, officially called Operation Chess,2 has primarily been limited to missile and air strikes targeting Iranian facilities in Syria as well as those of its proxies, especially Hezbollah. These attacks were not officially acknowledged before 2017. Israel has also carried out air strikes in Syria to disrupt weapons shipments to Hezbollah.
A more detailed list of Israeli actions in Iran–Israel conflict during the Syrian civil war:
By early December 2017, the Israeli Air Force confirmed at least 100 attacks over the past six years in Syria, all targeting arms convoys of Hezbollah and the Ba'athists. In September 2018, the Israeli Air Force stated that it had conducted over 200 airstrikes on Iranian targets in 2017–2018 alone.
Remarks
Due to some users indulging in equivocation, based on various interpretations of word allow, I specify what is meant here:
ALLOW
to give permission for someone to do something, or to not prevent something from happening
It is the second meaning that is used here: Israel actively works to prevent Hezbollah and more important actors from acquiring advanced weaponry, using military and diplomatic means. In some cases it is unable to do so - notably Hamas and Hezbollah were able to amass rather important arsenals of weapons due to Israel's reluctance to engage in a massive military action, like the one that we are witnessing for the last year.
As the commentators correctly pointed out, this is not without precedent - e.g., between the two world wars Britain and France were reluctant to enforce the terms of Versailles treaty, by allowing Nazi Germany to rearm, reoccupy demilitarized zones, and eventually betraying their allies (Czechoslovakia, Poland, and eventually France) in a hope to avoid fighting a war, which would have been constly in financial and human terms, and hugely unpopular - Indeed, upon return from Munich, both Chemberlain and Daladier received a heros' welcome in their respective countries... for having sacrificed the system of collective European defense against Germany (see Peace for our time).
