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5 of the TT is ^b6, so if you have ^6 in your melody, you will have a half-step dissonance that may cause issues.
Same with 2 of the TT being ^#2 (b3), which could conflict with ^2 or ^3, or 6 of TT being ^#6 (b7) and conflicting with ^6.
The common tones between two chords may not be enough to justify a substitution.
For instance, if you were to choose to substitute iii for I, then you will want to pay attention to whether or not the tonic is in the melody since that is the one note that is not shared between the two chords.
Are you trying to maintain strict counterpoint, just reasonably consistent voice leading, or just keep it sounding fairly consonant behind the the screaming Gilmouresque Strat/bebop saxophone/ambient Frippertronics solo you already recorded over the unmodified progression?
In a Jazz setting, this is often resolved by using an Altered Dominant chord, which would include all the notes from the major scale except the tonic, which isn't usually a melodic note on a dominant chord.
The 5 of the iii chord is ^7 (scale degree 7), a half-step below the tonic, so if the melody lands on the tonic during the iii chord, it can cause a lot of tension, possibly beyond what you may deem appropriate or would be considered appropriate for a given genre/style.
Similarly, you can look at the tritone substitution (which I'll call "TT" moving forward) and find that certain notes may cause issues like this.
With these notes in common, the two chords can fulfill the same function but there are differences between these chords, one of which being texture.
When choosing a substitution, it's important to pay attention to what the melody and other important parts of the arrangement are doing.