Tvheadend

First thing I'd suggest is watching the console messages from an esx console. (if your linux does X, I think console 5 or 6 might be the console messages, you'll have to experiment with that)
If you can see what causes the linux to go boom, might have a chance at sorting out the next level down.

Can you post (via pastebin or similar) the content of dmesg and /var/log/syslog? What exact version of ESXi are you running (inc. build number)?

I spent a great deal of time trying to get a VM on ESXi (5.0 & 5.1) to run TVH using a variety of Ubuntu setups (never with a GUI). The short version of my experience is that I now run Ubuntu natively on a HP MicroServer.

TVH does not deal well with an input device disappearing. I would suspect that the USB device becomes disconnected from the VM and TVH stops working.

Different people have different experiences of USB passthrough with ESXi. I was able to get it to work using a fairly high spec server with VT-d support, fast processor, etc. What I was not able to do was get TVH to work reliably (by which I mean for more than a few days without requiring a restart).

I tried using a variety of add in PCI & PCIe USB cards both by USB and VT-d passthrough to the VM. This proved very unreliable.

The PCTV Systems DVB-T2 290e generally worked well if it worked at all after the VM booted. I did ensure that the TVH VM had high priority for all resources and ring fenced things like memory. Beyond that there was nothing I did beyond using ESXi to passthough the device to the VM. I did find that I had to cold boot devices (power off VM / disconnect & reconnect USB device physically / power on VM) quite often. dmesg should show messages about the card being setup. If you can see the card in TVH then that part has probably worked OK. The next bit comes down to whether you get lucky with the hardware and the specific version of ESXi you're running. There are no supported configurations!

I would honestly suggest giving up and moving to a physical setup. I think I spent well over six months trying with ESXi and now I run a physical backend media server with a very large ZFS backed storage system with a reliable TVH installation using a mix of DVB-T / DVB-T2 / DVB-S2 USB inputs. It just works...

You could always then try Zen / KVM to run some VMs.

While not tvheadend, I've had good luck with ESXi and mythTV.
I'm using an HDhomerun for OTA and a Hauppauge HD PVR for Sat TV and have had no issues
since May of this year. Hasn't had huge load (2 programs at once every day.)
The 'server' isn't much of anything, just a generic PC with intel CPU. (so I can run Mac OS VMs)
Never had an issue with the HDPVR vanishing from the USB bus, just seems to work.

I've not used either of those devices but neither of them is a USB DVB tuner is it?

Using a HDHomeRun with a VM avoids any USB connectivity and might be a good option for some; not sure how TVH plays with one though.

Does HDPVR present as a tuner in Ubuntu which TVH could use?

The HDPVR is a USB video capture card(box?)
It captures from component, up to 1080i so throws a fair bit of bandwidth down the USB bus.
No, not a tuner to mythTV, just capture with an external script to change channels on the Sat receiver (IR blaster using lircd)
The HDhomerun works nice with tvheadend, I've got that working now, have been looking at tvheadend as a simpler solution over mythTV. (but haven't gotten word yet that the HDPVR works with TVheadend, it doesn't see it though)

Thanks Jon Boy

I think I have been slowly coming to this conclusion also and think its sound advice. The reason I want to stay with ESXI is that I work in IT and its good to have a home 'lab' for messing around with stuff. But as this is in 'production' for my home set-up I need something that is going to work reliably.

I will install Ubuntu Server and install Virtual Box Headless for my VM's that I need. That way like you said, TVHeadend and the DVB-T USB stick can run on the physical hardware instead of through a VM. Hopefully this will sort everything out. I'm hoping to do this, this weekend but will have to plan it around the Mrs being out of the house

I will let you know how I get on - Appreciate you sharing your experiencing and saving me allot of time