First, it is worth addressing what kind of tournament this is. If this is something like the Super Bowl, where the teams are generally widely-known year after year, then the answer will surely vary depending on the context.
Most long-time sports fans have a subjective preference for the other teams, ranging from 'second-favorite' to 'most hated', and these preferences are built up over time and are determined by the individual as well as the team. After having lost the Super Bowl XLV, a Seahawks fan may hold resentment towards the Steelers for years to come, and thus may root against them on all accounts.
On a more individual basis, fans may view another team favorably for a variety of personal reasons (e.g. a parent or close friend rooting for that team), and thus choose to root for them even if it came down to them beating their team. One writer even analyzed this idea in a FiveThirtyEight blog post.
However, if we were to remove these factors entirely -- say, we have a tournament where there is no prior opinion towards the teams that had advanced -- then the answer may shed light on both individual differences and group dynamics.
Studies have shown that the more a fan identifies with their team, the more loyalty they show to their team in the face of a loss:
When presented with any sort of negative information regarding the group,
highly identified group members react differently than those with lower levels of
identification (Cohen & Garcia, 2005; Ellemers, Spears, & Doosje, 2002). Highly
identified members typically reaffirm their group membership, while those with
lower levels of identification tend to distance themselves (Cohen & Garcia, 2005).
Such behaviors are apparent in sport fans. Wann and Branscombe (1990) found
that highly identified fans are less likely to distance themselves from the team
when the team lost than those who were less identified. Further, highly identified
fans exhibit biased attribution processing favoring their team (Wann & Dolan,
1994). That is, facing a win, highly identified fans ascribe the victory to internal
factors such as the skill of the team, or the coaching, and sometimes even fan support.
However, upon a loss, rather than conceding another team’s superiority, they
blame the loss on more external factors such as fate or poor refereeing. Thus,
highly identified sports fans seem to undergo some sort of biased attributional
processes when dealing with a loss.
And another study showed that the stronger the allegiance to the team, the more likely the fan was to perceive the win as a personal victory, thereby influencing their own self-efficacy. By contrast, the likelihood increased that a loss would be perceived as a personal offense.
Ss watched a live basketball game; then, in the context of a 2nd, unrelated experiment, Ss estimated their own performance at several tasks. Results indicated that fans' mood and self-esteem were affected by game outcome. More important, fans' estimates of both the team's and their own future performance were significantly better in the win than in the loss condition.
Whereas on the other hand, fans who identify less with their team tend to distance themselves from the team in the face of a loss.
So it could be the case that personal allegiance to a team determines how a fan responds to a loss. If the fan identifies strongly with the losing team, they may be less likely to view the winning team (opponent) positively, and thus they may demonstrate a greater tendency to root against that team in the next round. On the other hand, less team allegiance may implicate a desire to identify with the winning team, in which case the "we lost to the best" mindset may occur.