I would recommend keeping those old results around for at least a few years, since it isn't usually a big problem to do so. Having old records around to compare results to might be beneficial.
In many countries, medical professionals or labs might be required by law to keep your records around and hand those results over to you or any doctor you authorize for a number of years. For example, here in Germany, they need to keep normal laboratory results for ten years, for some other items this can be up to 30 years. Similar laws exist in the UK and the US (where the exact legislation is state-specific, as far as I can tell). Since it can be a hassle getting them when you, for example, move, I'd also keep copies myself, though.
The point of keeping old non-significant test results around can be:
- to see that a significant result later is a new addition. As such, if a later test shows elevated blood sugar / hemoglobin, a doctor at least has an estimate of when this problem started.
- to reveal a trend, for example decreasing Vitamin D levels making supplementation desirable, or slowly increasing TSH levels pointing to a beginning hypothyroidism. More tests can then be ordered if necessary.
- since "normal ranges" are just normal for a population, and not necessarily for you (see How reference ranges are established), even a value that is marked as non-significant might not actually be normal for you. Having many values from the past can show an outlier that would be within the reference range, but is a significant result for you.
One example where a trend within "normal" levels could be of diagnostic value is hemoglobin:
Paying close attention to routine blood test results can be an effective screening system for colon cancer which, when diagnosed early enough, can be treated effectively. A new study shows that most patients with colon cancer have a history of consistently declining hemoglobin levels up to four years before being diagnosed with the disease
Is your hemoglobin 'trending'? Routine blood tests may provide early warning for colorectal cancer
Since for most people this would just amount to asking for a copy at a doctor's visit and then putting it in a folder when coming home, keeping the records doesn't appear too difficult to not do it.