Should I look at the data of an experiment before the dataset is complete? For a research internship I am running psychological experiments online. As it takes a while until the experiment is done (meaning that enough people participated, so that a sufficient sample size is reached), I could already look at the data and run a few analyses with that incomplete data set. This way I could already see a trend in what the final result might be.

Are there any methodological reasons that would speak against that? For example, am I biasing myself in this way? Or is there any other reason this could be considered bad research practice?
Short answer
From an ethical standpoint, not including interim evaluations may be bad practice.

Background
I will start off with a more extreme case than in your question example, just for illustrative purposes, namely that of a clinical intervention study. If it appears that the treatment group (say, experimental medicine Y, instead of the standard of care treatment X) features substantially, if not significantly, more cases of serious adverse advents, or even deaths, that may (or may not) be related to treatment Y, the ethical best thing to do is put the study on hold until things are sorted out. This, to prevent any possibility of further physical harm caused by the experimental treatment. This happens quite regularly and should be followed by a report paper stating the results and a discussion on the best way to proceed this research, if applicable.

In a more experimental setting it may also be ethically best practice to evaluate preliminary data, as possible experimental design flaws, unexpected results (weird outcomes or artifacts in, say, left-handed people) or confounding factors may become apparent, and timely adjustments to the experimental protocol can be made. Why is this ethically correct? Because you may be subjecting people to a flawed experimental paradigm and wasting many hours of otherwise more productive time.

The case provided by the other answerer, where the experiment is stopped, while prior statistical power analysis was done, that is malpractice. Conversely, adding more subjects post hoc based on 'near-significance' is also questionable practice. But this is more related to what you do with the experimental interim data. In my opinion, they should be critically evaluated, but not so much on effect size, but rather on feasibility, correctness and validity - basically to sanity-check the study proceedings.