Is there a limit on the message size for SHA-256?

There is technically a limit, but it's quite large. The padding scheme used for SHA-256 requires that the size of the input (in bits) be expressed as a 64-bit number. Therefore, the maximum size is (2⁶⁴-1)/8 bytes ~= 2'091'752 terabytes.

That renders the limit almost entirely theoretical, not practical.

Most people don't have the storage for nearly that much data anyway, but even if they did, processing it all serially to produce a single hash would take an amount of time most would consider prohibitive.

A quick back-of-the-envelope kind of calculation indicates that even with the fastest enterprise SSDs currently¹ listed on Tom's hardware, and striping them 16 wide to improve bandwidth, just reading that quantity of data would still take about 220 years.

^{1. As of April 2016.}

There is no such limit, other than the maximum message size of 2⁶⁴-1 bits. SHA2 is frequently used to generate hashes for executables, which tend to be much larger than a few dozen bytes.

The upper limit is given in the NIST Standard FIPS 180-4. The reason for the upper limit is the padding scheme to countermeasure against the MOV attack that Merkle-Damgard construction's artifact. The message length l is lastly appended to the message during padding.

Then append the 64-bit block that is equal to the number l expressed using a binary representation

Therefore by the NIST standard, the maximum file size can be hashed with SHA-256 is 2^64-1 in bits ( approx 2.305 exabytes - that is close to the lower range of the estimated NSA's data center in UTAH, so you don't need to worry).

NIST enables the hash of the size zero message. Therefore the message length starts from 0 to 2^64-1.

If you need to hash files larger than 2^64-1 then either use SHA-512 which has 2^128-1 limit or use SHA3 which has no limit.