OptiPlex 3040 – Compatibility with Xeon E3-1270 v5

3040 has H110 chipset. no user has uploaded benchmark of any xeon cpu in this model.

its high end cousin 7040 which has Q170 chipset has user validated evidence of xeon support:

PS 5040 (Q170 too) has no evidence.

1) Xeon E3-1270 v5 has 80 Watts TDP (reaching much higher peaks in turbo boost mode), Optiplex 3040 designed with maximum of 65W in mind (I also haven't formally seen "K" version support in specification).
Generally you can fry circuits by excessive amperes flowing via conductors (incl. Voltage Regulator, Power Supply as already mentioned by @bradthetechnut ) which were not designed dissipate more power, especially considering the fact what 3040 is a minimal and cheapest configuration (incl. electronic components-wise) vs other higher-end models (Optiplex 7040).

2) Apart from mechanical compatibility there's also low-level BIOS detection to match CPU microcode (as well as PWM system dependencies for fan control and performance-state voltages). If microcode of your CPU is not in a list coded into BIOS - then CPU is very unlikely to function correctly (and Dell doesn't include CPU microcodes just for fun, they're quite restrictive here actually, and reluctant, it is also related to quality control/testing and warranties, as well as marketing and commercial drive to buy a newer models of course), may be even by not supplying higher performance-state voltages to benefit from better CPU (just a guess).

3) Some operating system do have additional licensing checks vs CPU, e.g. licensing logic on Windows 10/11 Home (and Pro) normally won't support Xeon and will complain about the need to get "Windows for Workstations" license, generally invalidating your OEM BIOS built-in windows-key.

In short - in might work but I wouldn't risk it (unless it's experimental curiosity with components ready to be wasted). It might happen outright or after some time (due to usage above designed thresholds under heavier than expected stress).