As noted in the comments, specific future events are impossible to predict; human behavior is a chaotic system.
That said, proceeding from our current working understanding of how representative democracies function, a handful of general trends may be predicted with reasonable levels of confidence.
1. More Extreme Electeds
Higher levels of polarization, defined as a stronger (more measurable) effect of party identification and shrinking of the unaligned center, results in conditions favorable for ideological extremes to survive primary elections and prevail in the general election in those districts where one party or the other holds a powerful majority (the so-called 'red' and 'blue' districts). As these districts increase in number (due to the shrinking population of unaffiliated voters as well as migration of citizens to geographies where they feel more socially comfortable), holding the proportion of these districts that produce extremists constant, we would expect to see a rise in the absolute number of extremist members of Congress proportionally.
2. Congressional Deadlock
A knock-on effect to #1, as Congress itself comes to contain more and more extremists (on both sides), the ability of party leadership to achieve even modest bi-partisan support becomes diminished. Highly polarized voters demand resistance of opposition agendas and uncompromising support for their own agendas, creating an environment where the incentive structure for those seeking office favors more vigorous resistance of policy actions that do not originate within their own party. This reduction in bipartisanship makes achieving majorities on middle-of-the-road policy objectives more difficult, and consequently party leadership must increasingly make concessions to its more extreme members to secure their support - often at the cost of moderate members. Not that Congress has typically been viewed as 'speedy' but a general lack of ability to get things done would be reasonably expectable. Because of how SCOTUS' members are appointed, this can occur within that body as well.
3. Shutdowns, Missed Deadlines, Dysfunction.
If #2 continues for long enough, or as polarization becomes more extreme without a clear dominance of one part over the other, it will eventually spread to budgets and re-authorizations which the Executive branch requires in order to carry out the day-to-day business of running the government. Parts of programs can cease to function, others can be rendered effectively moot.
4. Control Over Congress Becomes All-Or-Nothing
Again, as the phenomenon exists for longer periods of time or becomes sufficiently severe, the deadlock and disfunction can become bad enough that the only way to effectively govern is to adopt all-or-nothing approaches where achieving majority status at any cost is the only way to make meaningful progress on policy while faced by the demands of voters. In reality, this is probably the limit of plausibility because dysfunctional governments will eventually irritate their constituencies to the point that some degree of bipartisanship becomes necessary for reelection to be meaningfully possible for an incumbent.
Any, all, or none of these could actually occur, but these are what theory of representation says is more likely under high levels of, or increasing voter polarization than otherwise.
