The answer to your question is other questions: Is the existing liner (the orange terracotta) cracked? Do the mortar joints contain holes? Your picture is beautiful, but it is not good enough for me to perform a thorough chimney inspection. Therefore, the image is not good enough for me to answer the question. You can, with understanding, inspect the chimney properly yourself.
Understand the issue with chimneys and why liners are required...
Burning any carbon fuel creates carbon monoxide and water. Everyone knows about the deadly carbon monoxide, but most do not consider the seemingly innocent water. Since the burning is hot, the water will be in the form of a vapor and is expected to move up the chimney with the other gasses.
Natural gas (propane) burns at a very low temperature compared to other carbon fuels. So low a temperature that some water condenses as it moves up the chimney, especially when the chimney is cold. The more efficient the burner, the cooler the exhaust gasses, the more water condenses. In time, it was discovered that the condensate ate the mortar joints in the chimney which created carbon monoxide leaks into the interior of the home.
Ask yourself the following questions: Was this chimney, or any chimney directly attached to it, ever used to exhaust a gas appliance? If not, then the chimney is much more likely to be fine. If yes, then the chimney is probably shot, especially if the heater was not stone age. Those pilot lights did a decent job of keeping warm air flowing up the chimney and keeping it dry.
Sending a video camera down and inspecting the mortar joints can be done with any modern cell phone and a long stick (a painting pole would be nice). An expensive remote real-time viewer would, of course, be better. Consider where the chimney shares a wall with the interior. Check for heat and/or black marking (shadowing) on the inside of your home.
In short: Inspect your chimney inside and out. If it is clear of obstruction and in lacking no holes, then it will be fine. If you look at the walls and mortar joints of your chimney and find open cracks, then you will not be fine.