I'm not sure whether this will really solve your specific problem, but it's worth a try to update the NIC firmware and driver.
Seems like this is a HP NC 331FLR NIC (DL gen8 default 4-port NIC with the BCM5719 chip).
There's no update binary you can run from 4.1, but you can update all firmware components with the current HP Service Pack for Proliant image:
Or boot the server into a live-Linux of your choice and use the Linux update binary:
http://www.hp.com/swpublishing/MTX-ec0e18db6a8e4d978b57aa95d1
These will update the 331FLR NIC to Boot Code version 1.37/NCSI 1.2.37.
Then update the tg3 driver in ESXi with this offline bundle to 3.129d.v40.1:
You need the offline bundle file (BCM-tg3-3.129d.v40.1-offline_bundle-1033618.zip) from this package. You can import that in the vCenter update manager for easier deployment or (probably) install it from the ESXi shell with esxupdate --bundle=/tmp/BCM-tg3-3.129d.v40.1-offline_bundle-1033618.zip
I'm a bit rusty in the ESXi 4.1 CLI department though, you might have to resort to the vihostupdate utility or with PowerCLI Install-VMHostPatch remotely: