The rules for shields specify that you lose their shield bonus to AC when you use them to attack (i.e. with a shield bash), and in the case of bucklers, when you use that hand to attack.
When your shield isn’t a buckler, things are very clear: you keep the shield bonus to AC. Quite simply, nothing says you wouldn’t. The armor spikes are definitely not the shield itself, and attacking with the shield itself is the only circumstance in which your attack can prevent you from getting the shield bonus to AC.
The case of a buckler is a little more complex, because they say that “if you use a weapon in your off hand, you don’t get the buckler’s AC bonus for the rest of the round.” However, the armor spikes rules are very particular: “You can also make a regular melee attack (or off-hand attack) with the spikes, and they count as a light weapon in this case.” Note that the statement is that it counts as a regular or off-hand attack—not that it is actually used in either hand. That is, spiked armor uses up your offhand attack, but it doesn’t actually use any hand. And the buckler is checking whether or not your hand is used for the attack, not whether you used your off-hand attack.
So the answer is yes, you can use armor spikes without interfering with your shield. You can also use armor spikes as an off-hand attack while using a two-handed weapon. The only thing you cannot do is attack with one weapon as your main attack, attack with some other weapon as your off-hand attack, and then also attack a third time, with your armor spikes.