Major prestige and legitimacy hit to the Qing, which is significant given the number of rebellions they were facing.
Assuming the British haven't just sacked Beijing for a laugh, there was probably something major to precipitate it - and this might mean it takes a bit longer for peace to be established, especially if hardliners in the Qing court dig in. Alternately, possibly the British are able to impose peace relatively speedily, but still with a lot of distrust and hostility with the Qing. Fallout from the loss of Beijing might also effect how the succession to Xianfeng shapes out - IOTL the conservative circle of regents he appointed were ousted by his concubine Cixi and his brothers Gong and Chun.
This is all significant because the British went almost immediately from fighting the Qing to helping them against the Taiping - in fact, they had fended off an initial (very tentative) Taiping advance on Shanghai in August 1860, at exactly the same time their army was overcoming the Qing's Taku forts. Over the next few years Anglo-French forces took an active part battling the Taiping in the environs of Shanghai, most notably fending off another attack on the city in early 1862. British officer Charles Gordon was appointed to lead the mercenary 'Ever Victorious Army' in late 1862.
The direct military contribution of Anglo-French efforts to the downfall of the Taiping has probably been overstated, but they contributed in other concrete ways - e.g. in early 1861 the British envoy Harry Parkes had convinced the Taiping 'Brave King' Chen Yucheng from attacking the poorly defended city of Hankow, which had the effect of blunting the Taiping's great western offensive. As a result, the Taiping were unable to break Zeng Guofan's siege of the strategically vital city of Anqing, which fell in September, clearing the way for the final siege of Nanking. In 1862 several thousand of Zeng Guofan's troops, under his protege Li Hongzhang were shipped downriver to Shanghai on British boats, allowing them to act as a pincer against the Taiping.
So, if the British are soured on the Qing somewhat and/or are still deeply engaged in the north dealing with the fallout of the sack of Beijing, it is possible the Taiping get a new lease on life - the maximalist success would be cornering and destroying Zeng Guofan, who got into pretty desperate straits in 1861 IOTL anyway, and for Taiping sympathisers to win over enough of the foreign establishment to allow them to waltz into Shanghai and establish friendly trade relations. A more limited success might see them driving Zeng Guofan away, perhaps all the way back to Hunan, and some sort of tense neutrality with the foreigners in Shanghai, which would at least allow the Taiping some breathing room to reconsolidate in their core area.
Should also consider what the fall of Beijing does to Zeng Guofan's war effort - major hit to morale, at the very least. Possibly disorder at court, receiving contradictory orders, perhaps some trying to recall his army northwards to support the court (which would be awkward because in was a provincial force grounded in the west, in Hunan). Alternately, if things get shambolic enough and he finds himself in control of one of the few decent government armies going around, could he possibly make a play for the throne? Could even make it an act of desperation - the Qing have disappeared up north and are not supporting him, so he wants something to enervate his army and give him some chance of overcoming the resurgent Taiping?
The Taiping, to be clear, are still absolutely bonkers and there's heaps the foreigners will still find really off-putting about them - but if the 2nd Opium War has gotten nasty enough for the British to sack Beijing possibly the Taiping become the least bad option, or at least have more room to maneuver as the British find themselves occupied in the north trying to navigate the fallout of sacking Beijing.
Severe attenuation of central power will also be highly beneficial for all the other rebellions going around - Panthay in Yunnan, Miao in Guizhou, the Nian (which ranged pretty widely in the area south of Beijing IOTL, so with complete royal control withdrawn from the city...), Dungan in the Northwest (which only kicked off in 1862 IOTL, but longstanding tensions) etc.