Keats' "Ode on Indolence" is a fitting place to start because it actually makes reference to an urn (and/or marble statues), providing an interesting comparison with "Ode on a Grecian Urn." In "Ode on a Grecian Urn," the speaker is troubled by confronting his mortality. In comparison, he contemplates pros and cons of the immortal (but immobile and no longer living) figures on the urn. In "Ode on Indolence," the speaker argues (to himself and to the images he imagines) that it is better to be lazy and thoughtless than it is to be troubled by things like Love, Ambition, and Poetry ("poesy"). In other words, if the speaker of "Indolence" were to happen upon the urn, he might look it over (maybe even multiple times) but he would walk away, arguing that it is too mentally troubling to contemplate such grandiose things as Love, Ambition, Poetry, and Mortality (from the "Grecian Urn"). The speaker in "Indolence" wants to avoid any anguish that the speaker in "Urn" suffers from.
While the "Urn" poem features a speaker confronting deep thoughts (with some anguish) and the "Indolence" speaker seeks to avoid them, the speaker in "Ode on Melancholy" seems to offer a middle ground or a solution. The speaker in "Melancholy," rather than being "teased" ("Urn") by notions of mortality and rather than being thoughtless ("Indolence") suggests that if one should experience anguish or melancholy, try to enjoy it. This speaker says "go not to the Lethe" (don't try to forget melancholy). Instead of mourning the fleeting nature of life and beauty, reflect on how the fleeting aspect makes these things more significant. "She dwells with Beauty--Beauty that must die;" in other words, if something lasted forever, it wouldn't be as special. Therefore, beauty and melancholy go together. There is something good to be gained from being sad/melancholy; the sadness implies a loss of love, desire, and so on. The sadness is the trace and/or memory of love, beauty, etc. In this sense, the speaker of "Melancholy" would tell the speaker of the "Urn" to appreciate the anguish of knowing mortality because it implies a desire/love of life. The "Melancholy" speaker would likewise encourage the speaker of "Indolence" to think the same; to embrace deep thoughts, regardless of the melancholy that might result. The melancholy is a by-product of losing something beautiful; therefore, it is tied to that beautiful thing. The speaker in "Melancholy" might be suggesting that one should think of melancholy as an elegy, a memory, a memorial celebration of the beautiful thing, whatever it may be.