To put it simple: with increase in bitcoin price the value of gas fee rises.
It is only true if fee size is the same. If technically Bitcoin developers can reduce size of transaction, with a same fee rate, transaction fee will become lower or the same even if Bitcoin price is higher.
Could it decline the profit in bitcoin investment to the extent when we are not able to pay the gas anymore?
It is always a problem if you have too small bitcoin capital. If you have a big capital enough, transaction fee will not be what you care. Like if you have $1000 in bitcoin, $1 or $5 will be transaction fee you can afford to spend. It is not a good transaction fee with many people, I know.
If you care about transaction fee, want to save it as most as possible. Use Segwit address (Bech32m, bc1q). Consolidate inputs when fee is cheap because later with less inputs as 1 input, your transaction will have smaller size that is helpful to save transaction fee
[1]. Make your transactions in weekends when fees are cheapest compare to other days each week.
[1] Transaction size calculators
https://jlopp.github.io/bitcoin-transaction-size-calculator/https://bitcoinops.org/en/tools/calc-size/