There's two ways to interpret this question, with different answers:

1. How did the proximity of Rus' territory in modern day Belarus to pre-medieval Lithuania (the Lithuanian tribes and the Lithuanian land confederation of the late 12th century) affect their development?

2. What was the role of the Rus' territories in modern day Belarus which fell under Lithuanian suzerainity before the 14th century in the Lithuanian state, how did they influence the politics of the state?

I'll try to answer both these interpretations.

1

Archeological and literary evidence alike shows that the advance of the East Slavs northwards to Baltic territory in the upper Dnieper-Nemunas region was as much violent as it was peaceful. According to Eugenijus Jovaiša (if I recall correctly - I don't have his Aisčiai series at hand), the ancestors of the Lithuanians specifically lived to the east of modern day Lithuania before the 5th-7th centuries, when they were pushed westwards by Slavic expansion and colonized eastern modern-day Lithuania (and also brought a warrior tradition with themselves due to fighting against the early Slavs). From written records, we know that Yaroslav the Wise waged war against the Yotvingians in 1038 and Lithuanians in 1040, turning them both into Kievan Rus' tributaries. The 11th century was also a period of Rus' western colonization - Brest (1017), Navahrudak (1044), Braslaw (1065), Minsk (1067) were established during this time, pushing westwards into formerly Baltic territory. This process was likely violent - Navahrudak, for example, began as a fortress, while eastern Lithuania is similarly rich in hillforts dated to this time period.

The balance of power between Rus' and the Lithuanians shifted in the 12th century. The Principality of Polotsk, which was the principality most actively participating in expansion into the Baltic lands (although their expansion generally followed the Daugava river, so somewhat to the north of Lithuania), was partitioned into three in 1101, Rus' itself largely disintegrated, and Lithuanians turned from tributaries to mercenaries. Having lived in close proximity to the Rus' for centuries and displaying relatively extensive military organization thanks to the conflicts against Rus' itself, they became allies for various Ruthenian principalities (in 1159, they are mentioned as allies of the Glebovich family in a struggle for the throne of Polotsk, in 1180 they're mentioned as allies of Polotsk in the siege of Drutsk, etc.). The situation changed drastically from 1183 onwards - it's still not exactly certain what happened in that time, but Lithuanian raids, and specifically independent raids, rather than as mercenaries for foreign powers, drastically escalated in size and number from this year onwards. In Lithuanian history, this is sometimes called the "Land Viking" period - by the time that the Brothers of the Sword, the future Livonian Order, landed in Riga in 1202, the Lithuanians were the greatest, most terrifying threat for other Baltic tribes on the Daugava, Pskov, Novgorod and much of Belarus.

It's safe to conclude that it was this long proximity between Lithuanians and the Kievan Rus' in modern day Belarus which ultimately led to the formation of the Grand Duchy of Lithuania, one of the last large medieval conqueror-empires in Europe. It was not an incident which began out of nowhere, but a natural development of centuries of contact, conflict, and favorable conditions in the region which allowed a Lithuanian conqueror-people to claim vast territory.

2

Since you specified before the 14th century, and western Belarus this mainly concerns Navahrudak and the principality of Polotsk. Navahrudak was one of the first Ruthenian cities which fell under direct Lithuanian control. How it fell under Lithuanian control is actually rather unclear - in 1238, it was mentioned as the lands of Prince Iziaslav, then, after a decade of silence from written sources, it is mentioned again as the domain of Duke Mindaugas of Lithuania, specifically granted to his son, Vaišvilkas/Vaišelga (who later converted to Orthodox Christianity, briefly became monarch, then abdicated and retired to a monastery). In Belarus, and east Slavic historiography in general, it has become common to label Navahrudak as Mindaugas' capital - because sources on his capital, if he even had any, is extremely scarce, Navahrudak, being a town of some respect which had fallen under Lithuanian rule, holding this position is reasonable to assume. His coronation as the only King of Lithuania in 1253, supposedly, took place there, and sometimes it's even used to cast doubt on the Baltic origin of the Lithuanian state.

Of course, being educated in the Lithuanian side of medieval Lithuania's history, I can't claim to be the most objective judge in this old debate, and it is not impossible that Mindaugas, a very recent convert to Christianity, would consider coronation in a Christian city (Navahrudak had a Catholic church in this time period, too) - however, for a supposed capital of Lithuania, historical sources are very harsh judges on Lithuanian rule in Navahrudak. The Halych–Volhynian Chronicle, which often writes on Navahrudak in this time period, states bluntly that Vaišelga acted as a tyrant and executed two to three people in the city square every day. The same Chronicle draws a clear distinction between Navahrudak and Lithuania when describing the construction of Lavrashev Monastery by Vaišelga, stating that it was built "between Navahrudak and Lithuania". This does not inspire confidence in the claim that it was a core city of the Lithuanian state, a capital where the King himself would reign and be crowned - quite the opposite, it paints the image of a periphery, conquered in recent times and treated as such.

Navahrudak would later serve as Vaišvilkas' base of power during the succession wars after Mindaugas' death, and was given away to Halych-Volhynia for their aid, before being reclaimed by Traidenis. It would grow in prominence in the later centuries - Vytautas the Great built the Navahrudak Castle and turned it into a center of Orthodox faith in Lithuania, for example.

Polotsk, on the other hand, was a very important actor in Lithuanian history even before the 13th century - as mentioned, it was an active pusher to subjugate and colonize the Baltic territories in the west. By the time that the Lithuanian state formed, however, it had long lost its importance and in fact ended up quickly dominated by its former tributaries. Throughout much of the 13th century, Polotsk was in the Lithuanian sphere of influence - sources record Tautvila, Gerdenis and Constantine, Lithuanian rulers who were allies (or, in Tautvila's case, opponents) of Mindaugas and ruled Polotsk as its princes in succession. Tautvila used Polotsk as his base of power during one of Lithuania's numerous civil wars, being a key player in a coalition of Lithuanian-ruled Polotsk, Halych-Volhynia and the Livonian Order that sought to overthrow Mindaugas and enthrone Tautvila as the next ruler of Lithuania (it ended with Mindaugas accepting Christianity under duress and thus knocking the Livonian Order out of the coalition, leading to its collapse). Tautvila also waged wars against Novgorod, Vladimir-Suzdal, and placed his son Constantine in power in Vitebsk - showing that as early as the mid-13th century, Lithuanian influence had grown to dominate more or less all of modern-day Belarus.

In conclusion, the Ruthenian lands in modern day Belarus were quite important in internal Lithuanian politics in the 13th century, often serving as power bases for rulers and as a medium of expansion, but it'd be a hard claim to say that they were on the same standing as "ethnic" Lithuanian territories to the West. Consistently, the Ruthenian lands were treated as conquered lands, handed over to sons or allies, and sometimes even handled brutally.