And His voice shook the earth then, but now He has promised, saying, “YET ONCE MORE I WILL SHAKE NOT ONLY THE EARTH, BUT ALSO THE HEAVEN.” 27This expression, “Yet once more,” denotes the removing of those things which can be shaken, as of created things, so that those things which cannot be shaken may remain. 28Therefore, since we receive a kingdom which cannot be shaken, let’s show gratitude, by which we may offer to God an acceptable service with reverence and awe; 29for our God is a consuming fire. Heb 12:26- NASB
The context of Hebrews 12 lends no correlation to the burning bush.
It does express a contrast between the OC and the New Covenant. The new will eventually consist of only spirit and no flesh - no more corruption, and no more sin. All that remains of a physical age will be removed along with every trace of the evil and darkness that filled and soiled it to the core.
God as 'a consuming fire' has a plan to cleanse the earth and even the memories of it. We will have a new earth - it's still an earth, just a new earth.
For behold, I will create new heavens and a new earth. The former things will not be remembered, nor will they come to mind. Is 65:17
Nothing will be left that is not supposed to be left. Nothing will survive the intense fury and heat of God's wrath on evil and its residue.
But the day of the Lord will come like a thief, in which the heavens will pass away with a roar and the elements will be destroyed with intense heat, and the earth and its works will be discovered (or burnt up). 2 Pet 3:10 NASB