We will start with some of the basic rules/opinions regarding bal tosif. Tosafot[1] asks how we can blow shofar both before and during Musaf on Rosh Hashana without violating bal tosif and answers that simply repeating a mitzva more times than necessary does not constitute bal tosif. The Rashba[2] writes that one does not violate bal tosif if the additional activity is mandated by Chazal.[3]
Many Acharonim compare the act of affixing two mezuzot due to a doubt to that of wearing two sets of tefillin due to a doubt. The latter practice is the subject of much discussion regarding bal tosif.
The Shulchan Aruch[4] rules that one who wants to don the “tefillin of Rabbeinu Tam” in addition to “the tefillin of Rashi”[5] should, if possible, don both pairs at the same time. To avoid bal tosif, the Shulchan Aruch requires that one have in mind that whichever set is really the correct tefillin should be for the mitzva and that the one that is in truth not the proper tefillin is “no more than straps.” Tosafot Yeshanim[6] maintains that adding to a mitzva merely for the purpose of eliminating doubt is not subject to bal tosif at all, but many authorities disagree with this claim.[7] The Tur[8] rejects the relevance of bal tosif more fundamentally, arguing that it applies specifically when one adds on to the basic description and character of the mitzva – for example, when one has five compartments in the tefillin – but it does not apply when one simply wears two separate pairs of normal tefillin. Many take issue with the Tur based on a gemara[9] that states that one who finds tefillin on Shabbat outside an eiruv and wants to wear them so that he can bring them to safety may not wear two pairs at a time, because of bal tosif (among other reasons).
While we cannot summarize all the discussion on the matter, we will mention that the Magen Avraham[10] writes that one may don two pairs of tefillin only if one of them is not kosher. The Mishna Berura[11] explains that the Shulchan Aruch’s case of donning the “tefillin of Rabbeinu Tam” in addition to “the tefillin of Rashi” is permitted only because it has two factors that minimize bal tosif: 1) The extra element (the incorrect tefillin) is separate from the required element (the correct tefillin);[12] 2) One of the entities (the pairs of tefillin) is unfit for the mitzva. Even in this case, one should intend that only one of the sets of tefillin (whichever is actually the correct one) is for the mitzva.
The Pitchei Teshuva[13] posits that one who places two mezuzot on the same doorpost violates bal tosif. Why would one do such a thing? Poskim discuss, for example, the case of one who rents an apartment from a Jew who is lax about mitzvot and who has put up a tiny mezuza case covered by paint, which the renter does not have permission to remove. In this case, the question is whether the renter may affix another one.
Regarding your question of putting mezuzot on two different doorposts, where only one doorpost can be obligated in a mezuza, there is a disagreement among the Acharonim. The Binyan Tzion[14] maintains that the mezuza that is on the wrong doorpost has no more halachic significance than the wrong pair of tefillin discussed above. The Shulchan Aruch’s idea of donning two pair of tefillin simultaneously can therefore be applied to putting mezuzot on the two questionable posts. The Maharam Shick,[15] however, argues that a kosher mezuza affixed to a doorpost – even if it is the doorpost that in truth does not need the mezuza – falls within the realm of the mitzva, making it subject to bal tosif when it is opposite a mezuza in the right place.
While there is no clear consensus among contemporary poskim,[16] the stronger opinion is to not sanction mezuzot on opposite doorposts, whether as a clear ruling[17] or as a practical preference.[18] In addition to formalistic bal tosif issues, it is problematic to create an odd-looking new phenomenon of two mezuzot, even if it is out of a desire for stringency/covering all bases (which itself is very often a two-edged sword).