Most of these tips are for certain scenarios but overall they should give you some tricks to deal with hard situations. They are in random order.

-Start with Crusader Aura on every map and switch to Concentration Aura few seconds before you reach your target.
-Use BoK and BoM macros above to constantly rebuff in combat. They are extremely good for covering HoS/HoF/HoP and increase stats / mana regeneration.
-Refrehs Beacon of Light on the nuketarget every time it is dispelled, it is your way for more Holy Power and that way to more mana effiency.
-Dispel CC-effects first from players who can also dispel/remove some other CC effects.
-Hand of Freedom: Save it for slows you cannot dispel, or if you know you wont be dispelling for few seconds (you need to heal hard). Remember you can use Hand of Freedom even while cycloned.
-When enemy has mage or a warlock and your team mate is taking heavy damage, make sure you wont be interrupted during casts. Line of Sight the mage (go behind corner) or fear warlocks pet and heal hard then -> if you get full time interrupt you can easily lose. Save Aura Mastery for situations when you cannot fear the pet or LoS the mage.
-After a long CC-chain on you and your partner is low hp, follow the priolist:
3 HP (Holy Power) Word of Glory > Infusion of Light procced Divine Light > Holy Shock > Flash of Light > Divine Light.
Pop healing CDs if you are not sure you can get him up before he dies.
-Pop your healing CDs (Guardian of Ancient Kings, Awenging Wrath, Divine Favor) in chain if somebody is getting nuked hard. If you cannot outheal the damage, pop some of them at the same time since they stack pretty nicely.
-If you get switched, pop Divine Protection instantly to reduce the initial damage -> they might switch away simply because they are feeling they are not doing that much damage on you. Popping Divine Protection early also ensures you can have it soon up again.
-Holy Radiance with Hand of Freedom is extremely powerfull. Use it to run around pilars for max effect. Best for situations when you need to LoS the enemy team or they damage you hard or when you know you wont reach your team mates range in time.
-Try to use your stun as early as you can: games rarely last under 1min/40sec. Never use it on a target with a dispeller on sight with no better things to do. Try to lure a player behind a pilar and stun him there, he will be out of the game for 6seconds or he will use trinket. Support your teammates CC-chains with stuns.
-Use Divine Plea when drinking or when nobody is taking damage. After patch(4.0.6) you probably want to protect Divine Plea from getting dispelled by going behind a pilar.
-When you face a new interrupter, shaman/DK/warrior/feral/paladin/rogue/mage/warlock it is good to try them out at start of the game. See how easy they go for your fakecasting, cast FoL for 0.2-0.3seconds to see if they interrupt. If not, repeat with 0.7-0.9 sec cast. If they still dont interrupt, then you have a hard opponent. You will have to improvize and watch your casts closely. Remember than even the worst interrupters will probably learn your fakecasting if its a long game, so you will have to fakecast different style as the enemy evolves.
-Drink as much and as often as you can, have 5-XX stacks of best water in your bags so that you wont run out during a long game. Remember to inform your partners that you are drinking so that they can play a bit more defensive. Blowing out partners defensive CDs might be worth it aswell, just dont drink too long or your partner(s) will die.
-Never leave arena game if one of your partners die. It is still doable, maybe unlikely but Ive calculated about (on my personal experience of course) 1% chance of 3v3 game to be a win even if losing a friendly player before enemy. In 5v5 the number is lot bigger, around 20%. Tell your partners to still try their best. 1v1 situation is easily doable, if its a healer vs. healer it will usually end up a ressurection battle (let him cast ress as long as your ress is finishing first). Ask your partners to not release spirit and not to press accept instantly after they get the ress.
-Hand of Sacrifice is one of the best defensive CDs you have. If you are fast you can cast it on a friendly player taking damage before sheep lands so that the sheep will break before it is dispelled. Use it a lot againts team which do not have a breakable CC, to reduce the damage on one player.
-Judgement: Try to get one casted as soon as you can when arena game starts: free 9% haste. Remember to refresh it in combat aswell, it reduces your global CD from 1.5sec (default) to 1.365sec (if you dont have haste from other sources).
-Stealth teams: Check the enemy team members who are not stealthed for their buffs: if there is Mark of the Wild, they have a druid. If they dont have Leader of the Pack its a resto/balance druid. If you see they have a feral on start but the buff dissapers, the feral is over 100yd away from the teammates (on the opposite side of the map). If they have 2 stealthers, most likely combination is druid+rogue but I suggest to check the buffs. Againts teams with rogue getting into combat before sap is crucial, rush to enemy with Crusader Aura + mount and use Judgement/ Hand of Reckoning to get in combat. If you get sapped anyway, have your partners run away from you with mounts.
-Get yourself familiar with tracking the movement of every player on the field. This way you can predict some switches and CCs, and easily counter them.
-You cannot bubble if interrupted, rather play it safe than die with bubble unused.
-Never yell at your teammates or rage about the game. The game changes but preferably your partners dont (their skill might change, but who they are). Being the one who always gets mad after losing makes you unwanted team mate.
-Get yourself familiar with other class mechanics and how they work, dueling zones outside SW and OG are great places to test out stuff you are not sure how they work. Knowing who can dispels what and how is important.
-Fear DK pets often to reduce the damage and to prevent the usage of Gnaw in good situations. Dispel your team mates pets and heal them.
-If you are going to get CCd, why not take the most of it and pop Divine Plea on start of it and then remove it at the end of it when you can heal again. Doesnt work for cyclones.
-Get familiar with the names of your team mates defensive abilities and try to be the one who is calling them out. If you are playing with melees, their tunnelvision doesnt allow them to see if you are CCd and cant probably use Shield Wall at the best time. It will feel like big responsibility at start but once you know when to call them it will benefict you and your team mate a lot, he can focus on tunneling and you can focus on healing.
-When somebody asks for dispel, freedom or BoP, give it to him as fast as you can. If you are CCd/ out of LoS / on CD inform him that you cannot give it to him. If you have some skill on CD and somebody asked for it, inform him how long the CD is (+-5secs).
-Your team mates might be tunnel visioning and might not notice when you are CCd, if you have spellalerter it is good to call when there is a CC landing on you even before it lands. They will get much broader view of the match by just listening to you.
-Do not trinket stuff that your team mates can dispel unless its absolute emergency. When you get CCd, check your teams player who can remove that CC if he is CCd / out of LoS / out of range and then consider trinketing it.