EVE New Citizens Q&A

Depends on the agent that gave you the mission.

Normal missions for normal agents, timeout = mission failed, you lose (instead of gaining) the small amount of standings that you would have gained by completing the mission, and you don't get the ISK and item rewards. The agent will continue to give you missions, and doing a couple successful ones will probably repair whatever standings you lost from failing that one.

Career agent missions don't expire, but if you fail the mission or decline it, that agent will cancel the whole rest of the chain and won't talk to you anymore. But you can go to a different school and do the Career missions there.

Sisters of EVE Blood Stained Stars (epic arc) mission, the whole arc is repeatable every 3 months, so if you fail you have to wait 3 months before trying again (failing one mission cancels the whole arc).

Some confusion here and Memphis may have added to it.
Standard agent missions level 1 to 5 do not "time out" in the traditional sense that you have failed them as Memphis indicates.
Standard agent missions will stay active in your journal for a long time, I personally have had them active for more than two weeks. There are two ways to check this, open your journal and simply look if the mission is still there it is still active. The other is try and pull a mission from the same agent, if you have a mission active they will tell you that you have to complete the current mission before they can give you another.

Standard agent payouts for mission completion come four ways. The first three are ISK, LP and standings and these will payout no matter how long it takes to complete the mission. The fourth is a "bonus" based on completing the mission within a specific amount of time. The amount of time you have to complete and still receive this "bonus" is what you see listed in the mission briefing from the agent.