I think you've noticed a slight ambiguity in the poem. That said, 'patience' is a characteristic of people, not objects, so a simple reading (certainly my initial thought) is that the lampholder is the person holding the lamp.

Can an inanimate object have the property of patience, except in a poetic sense? Is the adjective "poetic" enough for us to suspend our thought processes as we search amongst the multiple layers of meaning for threads of insight? To my eye, this verse implies a soul, a presence, a person holding the lamp - a person with patience - just as it implies that the flame offers a generous gift with its light. Yet at the some time both the flame and the lampholder are represented as objects. If we let drop our aversion to ambiguity, or even dualism, this may not be a problem for us. Perhaps, as we see light both as a wave and a particle and accept that, for now, we cannot reconcile this duality. perhaps the flame can represent both light and warmth, and a spirit that offers us those. Could a lampholder be the bearer of a flame, both as an object and as a spirit, a person holding an object? Elsewhere in the Stray Birds series, we have; "What is this unseen flame of darkness whose sparks are the stars?" I am no Tagore expert. I wonder whether these poems are better considered both one by one and as part of a series.? I don't know anything about Ying Ting Lin, but she or he has put together quite an interesting page on this, I think, at https://www.behance.net/gallery/37796861/The-Stray-Birds-by-Rabindranath-Tagore-16

I would say it is the lamp holder the base of the lamp, because it is a one line poem or quote. I agree with Shona it is teaching to be grateful for small things that go unnoticed. And the base of the lamp the lamp holder will be in the shade. Also very commonly in the English language we apply human characteristics to inanimate objects, which can not be done in other languages. A lamp holder may not be able to be patient in many languages the logic and grammar of the language does not allow it. But it is valid common and often used especially in song,verse and poems etc in the English language.

It means the lamp base. I think the poem is trying to teach an appreciation of all round gratitude. Be thankful for small things that can go unnoticed.