Aside from the general obligation to be mesamach choson v’kalah I doubt there is any actual source where chazal bring down that lifting the choson/kalah on chairs, tables etc. is simply an extension of the custom for some to hoist the kalah (not the Choson) on their shoulders and dance. We learn this in several places including Ketubot (17a)
רַב אַחָא מַרְכֵּיב לַהּ אַכַּתְפֵּיהּ וּמְרַקֵּד אָמְרִי לֵיהּ רַבָּנַן אֲנַן מַהוּ לְמִיעְבַּד הָכִי אֲמַר לְהוּ אִי דָּמְיָין עֲלַיְיכוּ כִּכְשׁוּרָא לְחַיֵּי וְאִי לָא לָא
Very loosely translated: Rav Acḥa would place the kallah on his shoulders and dance. The Rabbonim said to him: What is the psak…is it permitted for us to do so as well? Rav Acha said to them: If brides are comparable for you to a beam, fine, but if not, no, you may not.
Apparently this minhag was subsequently extended to the choson but some rabbonim felt the custom of having the brides legs dangling over someone’s shoulders and carried around the simcha hall of that era was undignified, untziniusdik and potentially problematic from the perspective of negiah etc. and chairs were added to the formula. (Nitei Gavriel, Nissuin, 41:14)