I am making a wizard who mostly uses the environment around him to attack, with spells like Animate Object. Currently I want to have him carry a spiked chain with him for his Animate Rope spell. I know that the spell specifically says the rope cannot be used to harm, and it never mentions chains as a target. Does this mean it won't work as a restraint, or can it still be used to tie up opponents, and whenever they struggle to escape they take 1d4 piercing damage?
The chain is certainly a rope-like object, and so can be the subject of an Animate Rope spell. However you cannot use Animate Rope to tie up a resisting opponent. A subdued opponent would be fine, but an opponent actively struggling and resisting cannot be restrained by the Animate Rope spell. Instead, what happens is the opponent becomes entangled.
What happens whilst entangled with a spiked chain isn't covered in the rules. A little common sense can be applied though. The rules do state no damage is done by the rope whilst entangling, as mentioned by @DanRasmussen, but that is most likely damage from constriction. It's not unreasonable to expect the chains to do damage whilst entangling a struggling opponent. Whilst this will be entirely up to your DM, if this was my call I would rule that the chain does damage on the initial entanglement, and then again on every subsequent failed escape attempt, or every burst attempt (successful or not).
Yes, I would argue that you can animate a chain, as it is "rope-like."
However, a spiked chain probably has a diameter of more than an inch. As a GM, I would consider the spikes when determining the overall diameter. This reduces the length of chain that can be animated.
Secondly, you cannot consider your chain a "rope" for the purpose of being animated, then argue that it is not a "rope" in order for it to do damage. When it says "the rope does not deal damage," however, the description is referring to the rope/chain being thrown near the opponent, since it must be within one foot in order to entangle. Afterwards throwing the chain near your opponent, you can force them to make a reflex save to avoid being entangled. After entangling, the spell allows for no extra damaging effects, tho your GM may rule that the spikes do indeed damage the entanglee.