Dell PowerEdge T310 won't POST after memory upgrade
Take a look at the technical data sheet for your memory.
Specifically, the HMT31GR7BFR4A you have is dual-rank, not quad-rank.
Your system probably does not support dual-ranked 8 GB for 32 GB total (assuming your manual is correct, as it's a bit different from this technical guidebook). You'd want HMT31GR7BFR8A, which is quad-ranked 8 GB.
Also note from the technical guidebook:
T310 supports x8 (data width) and does not support x4 and x16 DRAM on RDIMM
You're attempting to install 2Rx4. HMT31GR7BFR8A is 4Rx8 so I'd expect it to satisfy these requirements.
Of course, other manufacturers (e.g. Samsung) also make RDIMMs in that configuration.
Diagnostic lights 1 and 4 means system board failure. Either you have the incorrect RAM, A bad ram stick or a bad system board. I looked at the firmware updates and none address an issue with ram.
The manual excerpt you provided states Single-rank and dual-rank RDIMMs of sizes 2-GB, 4-GB and 8-GB are supported for a total of up to 24 GB.
Your modules are 8GB dual-rank RDIMMS, and exceed the 24GB maximum as per the manual. Try with 3 modules.