Here some answers for the case where you're processing your data in a single batch (as opposed to online/in real time). These methods will take all of your data into account, so local sample-to-sample fluctuations aren't a problem. These are very well established methods and it should be easy to find existing code.
If your signal is increasing/decreasing linearly with respect to time, then you can fit a line to the amplitude vs. time curve using linear regression. The slope of the line tells you the amount of the increase or decrease per unit time. Alternatively, you can calculate the Pearson correlation coefficient. Its value ranges from -1 for perfectly decreasing to 1 for perfectly increasing, with 0 meaning no dependence. Unlike slope, correlation would give you a measure of the degree of linear dependence between signal amplitude and time, but not the magnitude of the increase/decrease.
If your signal is increasing/decreasing monotonically with time but the relationship isn't necessarily linear, then you can calculate the Spearman rank correlation or Kendall's tau. Similar to the Pearson correlation, these statistics will measure how much your signal depends on time, ranging from -1 (perfectly decreasing) to 1 (perfectly increasing), with 0 meaning no dependence. Unlike Pearson correlation, the nature of the increase/decrease can be any monotonic function, not just a straight line (e.g. an exponential or logarithmic curve).
For any of these methods, you can do statistical tests to make more well-founded statements about the increase/decrease. The issue is that you're trying to estimate something about the underlying signal from a finite, noisy set of data points. For a given true/underlying signal, any statistic you compute will be different each time you run the measurement because the noise is different. So, say your statistic says the signal is increasing; is this because it's really increasing, or could it be because noise just made it look that way? Statistical tests will help answer that question. Things to look into here are 'confidence intervals' and 'statistical hypothesis testing' (i.e. 'p values'). Depending on your application and level of noise, statistical testing may or may not be necessary. There may be nuances if your noise itself has temporal correlations.