You can use (almost) all of your powers when disarmed and without implements.
ImplementDDI and WeaponDDI powers don't prevent you from using them if you lack the appropriate tool: they simply allow you to get a bonus when you have the appropriate tool.
When using an Implement (keyword) power you make an unmodified ability check (usually Wisdom-based for Implement Cleric powers) against the target defense (as per definition of attack roll).
When you hold a magical implement (e.g.: a holy symbol), you get to add the implement's enhancement bonus to both attack rolls and damage rolls with Implement powers (and a damage boost in case of critical hit).
Also note that you can take advantage from any implement you are proficient with (not only holy symbols). If your Cleric took the Arcane InitiateDDI feat, then he can use even magical staves, wands and orbs to improve his Cleric powers.
A non-magical implement does not provide any bonus (except for flavor or feat support) unless it is a superior implement. Superior implements cost more but provides a varied array of advantages (properties similar to those of weapons).
For example, the Astral SymbolDDI costs 18 gp but has the Distant (+2 range of ranged and area powers) and Energized (radiant) (+2/+3/+4 damage bonus with Radiant powers) properties. Its base version is non-magical, but it you can buy or create an enchanted version of it.
You gain proficiency with a single type of superior implement by taking the Superior Implement TrainingDDI feat.
Lack of proficiency with an implement you are using means that you gain no advantage from its superior or magical bonus and properties.
When using a Weapon (keyword) power while unarmed, you make an unmodified ability check (usually Strength-based for Weapon Cleric powers) against the target defense (usually AC).
However, unarmed attacks have no proficiency bonus, so you lack a thick +2 or +3 modifier.
When you wield a non-improvised weapon you get a proficiency bonus to attack rolls with Weapon powers.
When you wield a magical weapon, you also get the weapon's enchantment bonus to attack rolls and damage rolls with Weapon powers (and a damage boost in case of critical hit).
Lack of proficiency with a weapon you are wielding means that you don't apply the weapon's proficiency bonus and gain no advantage from its magical bonus and properties.
Some powers may require you to wield specific weapons or implements. In this case, they explicitly state this requirement in their Requirement line of the power block. This is usually found in Rogue powers, for example.
The inspiration for it, like for many other things, comes from Tolkien and The Lord of the Rings, specifically the part where Aragorn, Legolas and Gimli chase down the orcs that have taken Merry and Pippin. Here's the direct quote:
Only Legolas still stepped as lightly as ever, his feet hardly seeming to press the grass. leaving no footprints as he passed; but in the waybread of the Elves he found all the sustenance that he needed, and he could sleep, if sleep it could be called by Men, resting his mind in the strange paths of elvish dreams, even as he walked open-eyed in the light of this world.
— J.R.R. Tolkien, The Lord of the Rings, Book III, ch.2, "The Riders of Rohan"
I would guess that this, as well as some other quotes from the same chapter (like "As before Legolas was first afoot, if indeed he had ever slept", and other supporting quotes) are the inspiration for "elves don't need sleep" trope that D&D embraced
Best Answer
The D&D Cleric, apart from its Blackmoor origins as a vampire hunter (as noted here), is a mish-mash that has grown into a trope of its own.
The D&D cleric as a trope, encompasses:
The Cleric was one of the three Original D&D† classes; the other two were Fighting Man and Magic-User.
The Cleric as evolved to add priest of specific cults - the original presentation almost ignored the deity.
In fact, the original published presentation is short enough to quote:
(D&D Original Edition, 6th printing, Vol 1, page 7.)
So, we have hybridization to some degree directly in evidence
More evidence of mashup are the titles for the levels:
(D&D Original Edition, 6th printing, Vol 1, page 16.)
Acolyte, Vicar, Curate, Bishop: All used in western churches, including Roman Catholic, Church of England, and Lutheran (tho' not all Lutheran Synods).
Patriarch is used in the Roman Church for 4 particular Archbishops; its use is far more common in the Eastern Churches - The Eastern Rites of the Catholic Church and the Orthodox Churches. Orthodoxy also uses Acolyte, Priest, and Bishop.
Adept and Lama are borrowed from elsewhere -
Lama from the Tibetan... for Abbot.
Which is itself an interesting placement - Abbots rank below bishops in Orthodoxy and Catholicism - but in the Early Irish church, were the ones appointing bishops.
Notably missing are the subdeacon, deacon, archdeacon, dean, and archbishop of the Catholic and Orthodox churches, and the High Priest common in pagan traditions, with the gaps filled by other traditions' titles. A sure sign of mish-mash. (Evil High Priests are listed as anti-Clerics in Men and Magic, p. 34, and on p. 9 under alignment tables under Chaos)
Later editions, in addition to expanding the spell repertoire, add priests of specific mythoi, and priests of specific deities within a given mythos. They also provide alternate titles.
D&D has had an influence, however. Fighting priest-spellcasters have become an accepted trope in fantasy fiction, even tho' the majority of priests in the Sword and Sorcery genere were one of three facets: Bookworm preacher, spellcasting priest, or warrior monk; all three can be found in the Conan stories. The D&D cleric has migrated into the video games and into a few Manga and Anime series. It has infected later RPG's, as well - the Shugenja of L5R has more in common with D&D clerics than with Shinto or Buddhist monks.
† Ignoring supplements for the moment.