Will there be a point in the future where moving bitcoins to a new address will make them safer? OP_SCHNORRCHECKSIG:
Exploring Schnorr Signatures as an
Alternative to ECDSA for Bitcoin

I've noticed (e.g., the above article) that newer forms of encryption are being researched and developed for use in bitcoin. Is it possible that moving bitcoins to a new address using new encryption will make those stored bitcoins less likely to be lost than keeping them where they currently are?
Will there be a point in the future where moving bitcoins to a new address will make them safer?


That is already the case.


Using a 2-of-2 multisig address (with keys separated) is safer against theft than a normal P2PKH.
Using a 1-of-2 multisig address (with keys separated) is safer against accidental loss than a normal P2PKH.


If native P2WSH segwit addresses get deployed in the future (with 256-bit script hashes, instead of 160-bit P2SH script hashes), we'll also get a way that is more secure against exhaustive collision searches.


  I am noticing (e.g. the above article) that newer forms of encryption are being researched and developed for use in bitcoin.


I believe the term you're looking for is digital signatures rather than encryption. While some cryptographic algorithms (notably, RSA) can be used for both encryption and digital signing, this is an exception rather than a rule. Bitcoin's authentication used in its script system is currently based on ECDSA, which cannot be used for encryption. The only places in Bitcoin where encryption is used is locally in wallets for password protection, and possibly on the P2P protocol if BIP151 gets adopted. If one of these is actually what you're asking about, I misunderstood the question.

The linked paper suggests the use of Elliptic Curve Schnorr signatures, something I've been working on myself as well. They have various advantages, one of which is a theoretical proof for its security. ECDSA does not have this, though it's not considered to be an issue in practice.

What Schnorr mostly helps with is privacy and compactness. The most interesting idea is signature aggregation, which would allow us to combine all signatures in a transaction - whether they are from multisig, multiple inputs or whatever - into just a single one.

Regardless, the answer to your question is absolutely yes. In particular, if we'd ever want Bitcoin to remain safe in the presence of sufficiently powerful general-purpose quantum computers, we'll need to switch from elliptic curve cryptography to post-quantum cryptography, which uses entirely different private keys, public keys, and signatures, and thus addresses.