The Siege of Sarajevo lasted for more than three years, and the people inside were very much trapped. This was in 1992, at which time the UN had ~40,000 troops, which it had mobilized for the specific purpose of stabilizing the situation. My question is then: what were the main reasons for the UN not being able to intervene and lift the siege for such a long time, despite having the means to do at least something?
When looking at why the UN does (or usually does not) do things, it's important to know two things: only resolutions by the UN Security Council are binding, and all of the permanent members have veto power. The permanent members at the time were the People's Republic of China, France, Russia, UK, and the US. With the Cold War just having ended, this was a period of unusually strong international cooperation. Even so, resolutions have to be watered down so as not to go against the wishes of those five.
That's the best I've got for why the UN did what it did. What I can cover is what they resolved to do and how that turned out.
The UN was looking to negotiate a cease fire and bring the warring parties back to the negotiating table. While this was happening, they tried to reduce civilian casualties and war crimes. To carry this out, in Feb 1992 the UN Security Council Resolution 743 created the United Nations Protection Force (UNPROFOR) with about 40,000 personnel. In particular they make it clear that:
...in accordance with paragraph 1 of the United Nations peace-keeping plan, the Force should be an interim arrangement to create conditions of peace and security required for the negotiation of an overall settlement of the Yugoslav crisis;
They were not looking to get themselves involved in the fighting, nor to impose a new peace by force; that would go from peace keeping to war making. They weren't looking to be an occupying force nor enforce a Pax Romana.
Resolution 770 upped the ante by declaring the war in Bosnia a violation of Chapter VII of the UN Charter which prohibits UN members from attacking each other. A violation of Chapter VII allows the UN to use military force.
UNPROFOR's mission expanded over the coming years. It began by protecting Sarajevo airport to allow humanitarian shipments (Resolution 758). A few months later it added protecting and escorting humanitarian aid convoys. In April 1993 Resolution 819 added safe areas, eventually including Sarajevo, which UNPROFOR had to protect. Finally, they had to monitor cease fire agreements.
All sides, especially the Bosnian Serbs, interfered with this mission. Yet Sarajevo airport remained open and relief supplies got through.
However, the safe zones were not adequately protected. Bosnian Serbs continually tested the UN's resolve by taking UN hostages, seizing weapons caches, and brazenly ignoring UN demands. The most obvious violation was Sarajevo, still under siege. It all culminated in the Srebrenica massacre when thousands of Muslim men and boys were massacred while the UN troops on the ground were too few to stop them, bound by rules of engagement from calling in support, and overrun.
After this UN failure, NATO intervened. They had previously been enforcing no-fly zones and making limited, reactionary air strikes. Now they began Operation Deliberate Force to provide air support for UNPROFOR. In September 1995 NATO aircraft struck hundreds of Bosnian Serb targets including air defenses, armored vehicles, and artillery pieces.
After the peace agreement, UNPROFOR was relieved by IFOR, NATO's [Peace] Implementation Force sent to enforce the treaty consisting of 60,000 NATO soldiers. This was later replaced by a NATO-led Stabilization Force (SFOR) until 2004 when it was replaced (mostly in name) by EUFOR Althea.
My conclusion is the UN bureaucracy was not ready to conduct a military campaign against an adversary that was not playing by UN rules. In their attempts to avoid escalating the conflict they failed to give their troops the means to conduct their basic missions. The Bosnian Serbs prodded UNPROFOR in a series of escalating incidents, learned their limits, what lies they would believe, and how to tie them up diplomatically.
The safe zone massacres finally cut through international hand wringing and allowed NATO and the UN to provide their peace keeping forces with the clear ability to retaliate and enforce the UN's demands. It's a shame it took three years and tens of thousands of lives to work that out.
For exhaustive detail on the UN mission, read Study of the battle and siege of Sarajevo which is Annex VI of the Final report of the United Nations Commission of Experts established pursuant to security council resolution 780 (1992).
UNPROFOR and UNTAC: Lessons Learned as to Requirements for Planning, Training, and Equipment by John O. B. Sewall, M. Gen., USA (Ret.) Vice Director of the Institute for National Strategic Studies, National Defense University