If the filesystem is encrypted, then no, they couldn't, as far as I understand the arrangement on newer iPhones (models with fingerprint readers).
Or rather, they could copy the contents of the phone's storage and mount it on a computer of their choosing, but they wouldn't be able to read it because it is encrypted with a 256-bit AES key which is (presumed to be) beyond their ability to brute-force.
The original phone holds a copy of that key, of course, but it is stored in the "Trusted Enclave", a separate system-on-a-chip which keeps the key in its internal EEPROM and won't reveal it until it sees a correct user passcode. The Trusted Enclave itself implements a timeout after repeated wrong entries, so even transplanting it into a custom-made circuit wouldn't help you to brute-force the (weak, four-digit) passcode.
Whether the Trusted Enclave chip could be breached physically, I don't know, and I doubt that could be answered without public-domain information. It would be a formidable task, though, even if the chip's casing isn't tamper-proofed in any way. The components you'd need to connect to are made from a small number of molecules, and even if you had the time and money and equipment to do it there'd be a strong risk of damaging the chip and losing any possibility of recovering the data.
If this is related to the San Bernardino case currently in the news, the situation is a little different because that relates to an iPhone 5C, which does not have the Trusted Enclave. Presumably you could open the phone and read the whole system image from the flash memory into a virtual machine (which ignores any code-signing protection), and then hack the OS to allow the passcode to be brute-forced. That's still a pretty big job, but it's not unthinkable that the FBI could do it in exceptional cases. It would be much easier for Apple, but there is also a political dimension to (a) the FBI requesting that and (b) Apple resisting.