Suppose I sign a contract in front of witnesses, but then immediately tear it up.
Is the contract still binding?
The contract remains valid.
Most contracts don't need to be written at all. Even if yours did need to be written under the Statute of Frauds or some other law, you're saying that it was made in writing, even though you later tore it up.
The fact that the contract is missing or destroyed doesn't change the fact that it exists and obligates the parties; it just makes it harder to prove what it said.
The only documents that lose their validity when the original is physically destroyed sufficiently are:
Last Wills and Testaments (this can be overcome if it is shown that it was destroyed without an intent to revoke the Will, e.g., in a house fire that also killed the testator or testatrix).
Live checks not yet deposited with a financial institution.
The original is important, but less absolutely and irrevocably, in the case of:
Negotiable warehouse receipts.
Negotiable share certificates.
Unrecorded deeds of trusts and mortgages.
Passports and visas.
Obviously, from a practical perspective, it is easier to prove a contract if a signed original is available to introduce into evidence. But, for pretty much every other document, this is a question of proof and does not change substantive rights.
In the case of currency, live checks, and items 1-5 of the second list, the basic issue is that it is customary to transfer and assign the rights arising under these documents by transferring physical possession of them, without recording the transfer in a central registry. Proof that you once possessed of the document isn't convincing proof that you are still in possession of the original and haven't transferred it to someone else.
Items 1-5 are less problematic if destroyed than currency or live checks, because often there are other ways to prove ownership or the existence of an obligation that are convincing.
For example, one can prove a promissory note existed from payment and possible an agreement related to collateral security for it, so one need only disprove that there was an assignment of it which is a fairly uncommon transaction compared to a transfer of currency.
In the case of items 6-7 on the second list, the statute of frauds (requirement that something be in writing) is applied quite strictly and it can be hard to prove that you had the real document prior to its recording when it becomes notice to the world with or without the original. Even if you can prove you had the real document, you can't record it without the original and so cannot get the full rights that comes with those real property documents.
In the case of Wills, this is simply a matter of long standing tradition with the force of law, and there is no real solid substantive reason that it should be treated otherwise, other than the difficulty of proving which Will was real and which was the last one when the author is dead and can't clear up that point, and lots of people have large economic incentives to lie about the question. Somehow, however, what should have been issues of evidence were transmuted into issues of substantive law in this area of the law.
In the case of passports and visas, this is because immigration officials at a border and firms transporting people internationally will commonly require the original documents for international travel, although some exceptions are sometimes made. This has facilitated the device of controlling possession of someone's passport to prevent them from traveling internationally. These can be replaced, of course, but only very lenient officials will let you travel using only a copy of those documents and it will be very inconvenient to lose them.
Contracts and consent forms and powers of attorney, for example, do not appear on this list.
Signing a document is totally overrated. Is there an offer, is there consideration, is there a meeting of the minds, and was there acceptance? I assume that someone gave you a piece of paper with some mutual promises. You signing the contract is a form of proof of your acceptance. You signing a contract and then immediately ripping it up indicates that you did not accept the contract. If the offeror relied on your apparent acceptance to his detriment, in that 1 second between you signing and ripping it up, then you might be estopped from arguing that you had not accepted the offer.
Anticipatory breach of a contract could result in some award of damages. However, the damaged party has an obligation to mitigate any losses. Since the offeror has not suffered any actual damage by you repudiating the contract, there would be no point in him suing you.
The usual situation in which this occurs is where a contract existed where one party delivered immediately, and the other party has an ongoing obligation. The contract exists for the benefit of the party who still has to receive such compensation.
This party can usually decide to relinquish their claim. In these cases, the intentional physical destruction of the contract may very well be interpreted as such a decision.