What sort of role did the medieval and early-modern era fortifications and citadels that were part of cities in Europe play when those cities became contested during the war? Did they play any part in the defence of said city, and if so, how effective were they at holding back attacking forces?
(P.S mainly directed at the eastern front, and the urban combat in places such as Poznan.)
Fortifications and citadels had a remarkable role in the siege of Poznan and to a less extent in the case of Budapest. In the case of Poznan, the final german resistance stood in the citadell, it was hard to assault by infantry and armored vehicles, it took almost three weeks to the soviets to control all of it even with overwhelming forces. This efforts kept important units busy in the siege of the city, some argue that it was one of the reasons of the soviets not advancing to Berlin as soon as February 1945. In the case of Budapest, waffen SS troops resisted several assaults in the city citadel and the Buda casttle hill was the last position that the german held before trying to break out of the city.
But in most of cases I would remark urban combat in modern cities with concrete builginds as a much important factor in the war, with assaults on other cities being much bloodier in despite of the lack of fortifications and being the Poznan example an exception.
You specifically asked about the Eastern Front which is slightly outside my flair but that's ok, one of the most important actions involving a fortress was at Brest Litovsk in June 1941. Originally part of Poland it was designed to be able to face an assault from any direction but had been fundamentally undermined by the partition of Poland in 1939, the West facing defences were already in the hands of the Germans.
Seven battalions from the 6th and 42nd Soviet Rifle Division had been deployed as a garrison of the Citadel, it was estimated that it would take three hours to deploy them within the fortress in the event of an alert. Facing this force were nine German battalions with a further eighteen on their flanks, they would attack to the North and South of the Citadel whilst a large artillery bombardment pulverised the defenders. The general feeling was that the assault would be fairly easy, the first days objectives were some 5km to the East of Brest.
At 03:15 22nd June 1941 the attack begun with a bombardment that appeared to be overwhelming, Chaplin Rdolph Gschopf with the 45th Division recalled:
This all embracing gigantic barrage literally shook the earth. Great fountains of thick black smoke sprang up like mushrooms from the ground. As no counter fire was evident at that moment we thought everything in the citadel must already have been razed to the ground.
The Russians were taken completely by surprise and German troops crossing the river to attack the fort were not fired upon though some were killed by German artillery falling short. Almost all of the outer defences were swiftly captured, many defenders slept in outside tents during the summer and they were killed before they even knew the war had started.
However around the citadel itself the Russians began to resist more fiercely, at 10:50 the 45th Infantry Division sent to XII Corps Headquarters:
The Fighting for the Citadel is very hard - many losses
It was almost impossible to bring German reinforcements up to support the attacks and the resolve of the Russian defenders surprised them, after the fighting the Germans attributed many of the difficulties in taking the Citadel with the close quarter nature of the fighting which prevented them from employing heavy artillery in support of infantry attacks and the thickness of the fortress walls which rendered lighter weapons useless.
By mid afternoon the Germans had concluded that the fortress could not be taken by infantry assaults alone, they had suffered 311 killed in one day, two thirds of what the Division lost in France.
The 22nd of June was largely a day where the German attacks achieved tremendous success, the Russians were taken wholly by surprise and suffered tremendous losses though it is worth nothing that they recovered from the shock and began to fight back tenaciously against the Germans.
Brest Litovsk continued to resist into the second day. The Germans realised that their attempts to withdraw infantry to avoid losses had simply led to the Russians expanding their perimeter. Attempts to allow the Russians to surrender were partially successful, some 1900 defenders chose to do so but the fortress held on despite crippling water shortages and the deployment of more and more German infantry and artillery as well as more unusual weapons like French tanks and tear gas.
By the 30th June the Fortress had fallen to the Germans and was cleared of the few surviving defenders. Some 482 Germans were killed and 1000 more wounded compared to between 2000 and 3500 Russian dead.
Despite this seemingly final defeat there were continued incidents through July, the fortress had a system of underground tunnels and within them can be found crudely scratched messages "Things are difficult but we are not losing courage", "We die confidently, July 1941" and another proclaims "We die, but we defended ourselves. 20.7.41"