Depends on the keyword of the power used in summoning the creature.
If it has the conjuration keyword, the following probably exempts it from most auras/zones (emphasis mine):
Unaffected by the Environment:
Terrain & environmental phenomena have no effect on the conjuration. For example, a conjuration that is an icy hand functions in an inferno without penalty. The conjuration does not need to be supported by a solid surface, so it can float in the air.
Normally, a conjuration cannot be attacked or physically affected. If a conjuration can be attacked or physically affected, it uses your defenses. Unless an attack specifically targets conjurations, only the attack's damage (not including ongoing damage) affects the conjuration.
If the power has the summoning keyword, then those exemptions do not apply. The rules don't say anything about it having or not having a turn, merely that it has no actions of its own. Per the Rules Compendium p.189,
In a round every combatant takes a turn. (Thanks Simon Withers!)
This means that, by RAW, it must have a turn, although when in the initiative order its turn falls is left unspecified. There are 3 potential solutions to this issue:
1) The summoner's turn is also the summoned creature's turn, and thus anything that would affect it at the start of its turn triggers at the start of the summoner's turn. While the simplest approach, the risk here is that having it be two creatures' turns simultaneously might confuse certain timing rules, including certain effects that can trigger "once per turn" (thanks again Simon!).
2) Roll initiative when the creature is summoned to determine its place. Seems like a hassle.
3) Put the creature directly after its summoner in the initiative order. This has the advantage of being very easy and following the pattern established by creatures summoned by NPCs (thanks again Simon!). The only drawback to this choice is that it allows the summoner to ensure that the summoned creature always avoids any zones that would harm it at the start of its turn, since the summoner can move it out of those zones at the last minute (assuming they care).