You only need to actively gear yourself up before Wrath if your character is primarily a Healer, but you prefer to level up as Melee. For example, you're a Resto Druid but you level as Feral. Or you're a Resto Shaman and you prefer Enhancement over Elemental. If that's the case, you definitely want to put together a passable set. Dungeon blues, at least. You'll get upgrades to those blues pretty fast in Northrend, but you won't have an entire set thrown in your face the moment you install the expansion. You obviously need to at least be capable of fighting mobs as the spec you want to be, unless you're willing to suffer awhile as Elemental or Balance while you accrue enough quest rewards to be able to enter melee combat.
If you're okay with Balance or Elemental or whatever and your primary set of gear is Healing gear, you don't really need to worry unless you have nothing else to do but grind gear until the expansion. You have a couple of things going for you in Wrath that will help things out. The biggest one is the Spell Power change. All of that Healing gear you have will switch over to Spell Power, which will up your soloing power nicely. The rest of the stats on your gear may not be ideal, like a Resto Shaman may not have as much Crit on his Healing gear as he'd like on his Elemental gear. Not that exact scenario perhaps, it's just an example. Your Healing gear won't be the ideal DPS gear, but it will work. You do not need to panic.
Now, this doesn't mean you'd be wasting your time working on your gear before Wrath. Good gear will give you an edge for the first few levels. Your dungeon blues, and to a lesser extent Tier 4, will be replaced in the first couple of zones, yes. If you have easy access to Tier 5, though, that'll last you until the mid-zones. Tier 6 and above will last you almost the entire leveling process. If you don't have a raid yourself and you're so inclined, a lot of servers have guilds that sponsor Mount Hyjal PUGs for every boss minus Archimonde. That loot will last you until the high-70s, easily.
If you enjoy playing WoW, enjoy playing WoW. Don't get too down in the dumps that your gear will be worthless soon, because it won't be. Do you need amazing gear to level in Northrend? No. It will give you an edge though, if you're that kind of player. Uther knows I am.
To sum up: Healers, don't worry, your healing gear will be just fine as starter leveling gear in Northrend. If you feel inspired to get gear better suited to leveling, more power to you. It'll pay off.
That's it for this week, short and sweet. You get a break from my ridiculous chattiness. You can thank me whenever you'd like.