My question: Were the internal temp. of the EARTH to rise, would the resultant increase in heat radiating outward be significant to add to global warming?
Short answer: no.
Obviously, a full answer depends on just how much the internal (I presume you mean deep subterranean) temperature were to rise, and how much it would affect the average surface temperature, but “no” is a good answer for all realistic purposes. Keep in mind that the earth’s core is already about as hot as the surface of the sun. Volcanoes and such affect climate almost entirely through the emission of greenhouse gases and aerosols and therefore changing the balance of solar heat transmission and retention, and extremely little through direct thermal contribution, most of which escapes into space very quickly anyway.
That obviously depends on just how much the internal temperature would rise. But there’s no known mechanism by which the internal temperature could rise enough to cause a significant surface increase.
In the short term, any plausible increase in interior temperature would probably decrease the surface temperature, since it’d increase volcanic activity, and volcanic ash blocks sunlight.
Here’s a pretty good wiki: Earth's energy budget - Wikipedia
For round numbers geothermal energy is 1/40th of 1% of the total energy arriving at the Earth’s surface. The other 99.975% of the energy comes from the Sun.
So unless your internal temp increase is close to making the entire planet molten, the answer is pretty much “No.”
The internal temperature of the Earth is already above the melting point of Iron - how much hotter do you want it?
I’d say there is no known mechanism by which the internal temperature could rise at all. There are only 2 sources of heat - residual heat from when the planet first formed, and radioactive decay. I don’t see how either could increase.
Another way to think about it: caverns, until they get too deep, are cool. Good old Carlsbad is a steady 56 F.
I realize the OP specifically asked about heat radiating to the surface, but it got me wondering if it could result in more volcanic activity leading to great CO2 emissions?
Getting smacked by a big asteroid could increase internal temperatures … although such would be immaterial to what would be an extinct human species … and during the Sun’s red giant phase some heat energy could conduct down as the surface of the Earth was being vaporized …
But even a hundred cubic miles of 2,000 K ash dumped in the atmosphere by a volcano won’t effect temperature very much … maybe for a couple of years but the ash all settles out … it’s all the greenhouse gases emitted that stay in the atmosphere that will raise the temperature …
There’s also heat produced by tidal stresses from the Moon and Sun. Increasing either of those and leaving the Earth still inhabitable afterwards would be a bit tricky, though.
Although deep mines are hot. See Geothermal gradient - Wikipedia for more.
The short version is a couple hundred feet of rock insulate your cavern from the insolation and produce 50-ish temps in them.
A half km or more of depth take you down to where the internal heat predominates and it gets rapidly killingly hotter with increasing depth.
Carlsbad is cool at 56 F in the summer. But in the winter, it’s quite warm, at 56 F. It’s not that caves are cooler or warmer than the surface, but that they average out the surface temperatures.
Only somewhat related, but something like this is thought to occur on Venus.
One thing they noticed when they started mapping out the surface of Venus was that it didn’t have anywhere near enough craters. It’s like the entire surface of Venus is too new. One theory that has been put forth to explain this is that the heat inside Venus can’t easily escape like it can on Earth. Over time, the heat builds up and builds up until it gets so hot that the entire surface of the planet melts. Then the heat escapes, the surface cools, and the whole thing starts all over again.
On Earth, plate tectonics allows the surface bits to move around and let the heat out, so you never get this kind of heat buildup.
So the OP’s scenario can happen, just not on Earth.
Ah, but the OP didn’t specify the magnitude of the increase, so the answer is actually “yes”. 
So conduction isn’t fast enough to transport energy created in the interior, but convection is fast enough to cool it back down? Do you have a cite with some numbers, cause that sounds fascinating?
A massive lake of molten carbon the size of Mexico is discovered under the US, and it could cause climate CHAOS
"Melting carbon covers an area of 700,000 sq miles (1.8 million sq km)
Upper mantle could contain up to 100 trillion metric tonnes of melted carbon "
It’s carbonate, not carbon. Here’s the actual news release from the Univeristy of London
And it has nothing to do with climate change unless you’ve moved the goal posts to the +50 million years from now spot.
Well, to be fair, PCP gets all his (her?) facts from the Daily Mail, so you can’t really blame them for posting nonsense.
From Grey’s link.
“We might not think of the deep structure of the Earth as linked to climate change above us, but this discovery not only has implications for subterranean mapping but also for our future atmosphere,” concluded Dr Hier-Majumder, “For example, releasing only 1% of this CO2 into the atmosphere will be the equivalent of burning 2.3 trillion barrels of oil. The existence of such deep reservoirs show how important is the role of deep Earth in the global carbon cycle.”
PS - Which tabloid has the Page 3 girls?
Pearl Clutching Provocateur: Maybe you should state why you posted that link originally and what you think it means, and why you think it’s relevant to this conversation.
To the best of my knowledge the facts are these:
[ul]
[li]The Daily Mail has been a consistent source of fake news promoting shamelessly false facts to bolster climate change denialism for many years, but[/li][/ul]
[ul]
[li]In this case they don’t seem to have done much more than reprint a fairly standard press release, presumably because they believe that readers interested in denialism will read all the wrong things into it without needing to be prompted, and[/li][/ul]
[ul]
[li]These sorts of discoveries aren’t particularly new and have absolutely zero relevance to contemporary climate change discussions. If I had a dime for every time some legitimate research was misinterpreted in order to further climate change denialism, I’d be rich.[/li][/ul]
So again, my question is, why did you post that link in this thread? What does it add to the discussion?