No. A side channel attack is one that monitors a cryptographic process while it's operating, and uses leaked information to determine the secrets. Examples include: monitoring the CPU for RF emissions while it is encrypting, monitoring the power consumption of the CPU while it is encrypting, monitoring the time it takes to encrypt, monitoring the noise the CPU makes while it is encrypting. In every case, the encryption it is monitoring is already legitimately in possession of the key.
A powered off system that does not have the key will never perform a legitimate encryption/decryption operation. Therefore it has no attack surface for a side channel attack.
It is still susceptible to other attacks such as password guessing, forensically examining the swap file to see if the legitimate password was ever stored in memory that was swapped out, or threatening the former keyholder until he reveals the key. Just not side channel attacks.
A comment below asks if the attacker could extract the password from components.
Generally the key bits won't remain in dynamic electronic components for very long (exclusive of freezing the DRAM chips as @TomLeek points out.) Components such as bus latches, network adapters, etc., won't typically ever house the key, and even if they did, their memory buffers are generally very short, dynamic, and designed only for transient data.
Some people have theorized that a key stored for an extended duration in the same physical cells of RAM might cause a "burn-in" effect such that the memory cells could be differentiated with some specialized equipment or software. The theory is that if you knew which cells held the key bits, you could determine how long they can hold a charge before requiring refresh. The idea is that a bit that held a 1 might decay from a 1 state slower than from a 0 state. Such attacks have been demonstrated in a laboratory environment, and who knows, perhaps a government may be able to use them in a high profile spy case, but are not within the capabilities of an ordinary police investigation. Criminal organizations are not yet known to have these capabilities.
PGP Desktop marketing calls this effect "static ion migration" and attempts to mitigate this avenue of attack by keeping two copies of the key, and periodically inverting the key bits of each. (Ironically, it turns out to be a blessing for side channel attacks, as a RAM scraper can efficiently look for memory that is the bit inverse of another area of memory.)
See this paper, RAM is KEY for more on finding keys in volatile memory.
If the key was ever stored to permanent storage, such as HDD, flash, or CMOS, though, recovery becomes possible. The application has to keep track of every place it might have been written, and overwrite it. That's almost impossible with flash memory, because the wear leveling algorithms continually move where data is stored.