Carbon dioxide isn’t the only compound readily dissolving in water of any kind from the atmosphere. But it really isn’t of much concern. At first, remember that we are dealing with the following equilibrium:
$$\ce{CO2 (g) <=> CO2 (aq) <=>[+H2O][-H2O] H2CO3 (aq) <=> HCO3- (aq) + H+ (aq)}\tag{1}$$
Obviously, we cannot neglect most of this equilibrium – as you mentioned, the pH value does tend to decrease from 7 to around 5 or 6 if distilled water is left standing – but nevertheless the equilibrium is strongly shifted to the left, i.e. the solubility of carbon dioxide is relatively low and carbonic acid is a transient species for most intents and purposes.
Looking at the equilibrium we are dealing with also alerts us to the kinds of processes that could be affected by dissolved carbon dioxide. Evolution of carbon dioxide by some other reaction mechanism (e.g. oxidation of oxalic acid or oxalates with permanganate) would, for example, hardly be affected. In such a reaction, additional carbon dioxide is created which would shift equilibrium (1) away from $\ce{CO2 (aq)}$ – but since the right-hand half of the equation is already near saturation, the strongest effect will be a shift to the left, i.e. liberation of carbon dioxide from the solution, observed by characteristic bubbling. In a sense, using water that is already saturated with atmospheric carbon dioxide would be beneficial in this kind of reaction as carbon dioxide liberation would begin almost immediately.
There are two reactivities of carbon dioxide that might form a problem in certain contexts. The first one is the (weak) acidity of carbonic acid, which could lead to reactions with bases. This is important especially in titrations when a specific concentration of base is desired which should remain stable over time. If basic solutions (such as sodium hydroxide solutions) are left standing, they will not only absorb carbon dioxide but also react as shown in equation (2), leading to a reduction of basicity.
$$\ce{NaOH (aq) + H2CO3 (aq) -> NaHCO3 (aq) + H2O}\tag{2}$$
The second is carbon’s ability to react as an electrophile. This is essentially the reaction that generates carbonic acid from carbon dioxide (nucleophilic attack of water onto the central carbon atom, breaking one of the $\ce{C=O}$ double bonds), but water is not the only nucleophile one has to worry about. Admittedly, this is less of an issue in aquaeous solutions where water is just as good a nucleophile, but carbon dioxide can dissolve in practically any solvent and it can have greater effects in other solvents that are not nucleophilic. For example, Grignard reagents, which can be thought of as carbon-centred anions, can react with carbon dioxide as a nucleophile giving carboxylic acids; see equation (3).
$$\ce{EtMgBr + CO2 -> Et-COO- + Mg^2+ + Br-}\tag{3}$$
This reaction has synthetic utility if one wants to insert a carboxylic acid, but more typically one wants the Grignard reagent to undergo a different reaction and the attack would be considered an undesired side reaction. Grignard reagents themselves aren’t stable in water (as they are strongly basic, they would be protonated by water) but this can be an issue in ether or THF, solvents typically used for reactions with Grignard reagents.
The above notwithstanding, carbon dioxide is often the least of one’s problems in a lab setting. While many reagents and reaction conditions are incompatible with it, there are also many that are incompatible with oxygen gas, water or both. Most of the time, reactions in research labs will be run under inert gas (nitrogen or argon) mainly to exclude water and oxygen from interfering. Sometimes water is the major issue in which case oxygen exclusion often happens anyway as there is ample water in the atmosphere as well. These reactions are performed with dry solvents which already come packaged under inert atmosphere (often argon). Sometimes, only oxygen will have adverse effects in which case water (and all other solvents) will have to be degassed. Degassing will, as a side-effect, also remove carbon dioxide but oxygen will often be the prime target. Thus, not much thought is given to carbon dioxide specificially.
With advancing knowledge of reactions and mechanisms, it will become easy to predict where carbon dioxide might be a problem and where it isn’t – or where it is but so is e.g. oxygen – so you will find that it is rarely mentioned explicitly.