No, 5G will not replace LTE in the foreseeable future, it will largely enhance it.
For one, in the currently use frequency range, below 3 GHz, which is the only range suitable for covering larger areas, 5G is so similar to LTE, that it is barely distinguishable.
For the next few years, 5G will in general be a speed-boost enhancement for existing LTE networks in small, dense, areas (think shopping malls, football stadiums, and so on).
LTE networks will certainly be the main technology of cell phone access for the next 5 years, likely longer.
Even then, 5G will likely not replace LTE for major coverage cells in the next 10 years, though we will see a portion of lower-frequency spectrum (below 1 GHz) refarmed from LTE to 5G.
Contrary to medial reception, major 5G networks are still some years out and there is no point to worry about obsolescence.