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I fail to see why a tube tester needs precision resistors, especially a kit type. However, if you need the value to be as close as 1% you must decide what long term stability you need. Today's resistors are a lot better than the ones from the old days. In fact most resistors I have tested are very close to nominal value, while the old ones had more of a spread.
I'd suggest metal film but even deposited carbon may be stable enough. Wirewound are the most stable but aren't usually available in stock values, plus they have inductance that might present problems in some circuits.
Yes, precision resistors are less likely to drift because they are made of different materials that have been found to be more stable.
So my answer to your question at the end of the first paragraph is that it's okay to do that if you are using a modern resistor.
As for the burned resistor, I would suspect instead that the power rating is marginal, and if the tube is left drawing current for a long time it might get too hot. Calculate the dissipation if you can.
I assume you are measuring these resistors with a precision bridge, not just a digital Ohmmeter. For lower values you will need Kelvin connections.
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