I couldn't find any evidence to support either of these claims, so I decided to investigate.
I looked at five different methods of foundation application: my fingers, traditional wedge-shaped sponges, a Beauty Blender, a tapered foundation brush (specifically theBalm's Blend-a-Hand), and a buffing foundation brush (Tarte Airbrush Finish Bamboo Foundation Brush).
For my foundation, I used Maybelline Dream Liquid Mousse. I applied my foundation ten times for each method, and recorded the mass of the used product. For some reason, I thought applying and removing foundation 50 times would be some sort of zen experience, but it was not. It was boring as hell, even with a podcast on in the background.
Without further ado, here were my results, from most product used to least product used:
#5 (worst): Wedge Makeup Sponges
Raw Data (in g): 0.781; 0.512; 0.638; 0.470; 0.621; 0.797; 0.634; 0.793; 0.449; 0.364
Mean=0.6059, Standard Deviation=0.1544
#4: Tapered Foundation Brush
Raw Data (in g): 0.463; 0.345; 0.309; 0.304; 0.400; 0.208; 0.285; 0.285; 0.175; 0.139
Mean=0.2913, Standard Deviation=0.0991
#3: Beauty Blender
Raw Data (in g): 0.346; 0.442; 0.272; 0.333; 0.227; 0.266; 0.241; 0.162; 0.227; 0.205
Mean=0.2721, Standard Deviation=0.0815
#2: Buffing Foundation Brush
Raw Data (in g): 0.252; 0.250; 0.211; 0.207; 0.181; 0.151; 0.242; 0.216; 0.313; 0.308
Mean=0.2331, Standard Deviation=0.0512
#1 (best): Fingers
Raw Data (in g): 0.324; 0.141; 0.167; 0.196; 0.225; 0.211; 0.175; 0.212; 0.243; 0.173
Mean=0.2067, Standard Deviation=0.0512
|Hooray for scales.|
I did not compute a bunch of post-hoc tests to differentiate what was different from what, but I can tell you that the p-value was only marginally significant when the traditional sponges were removed from the equation (p=0.06). Thus, the big statistical conclusion was that wedge sponges are a shitty way to apply your makeup, but the other ways are much more similar. That said, this analysis indicates that if you are using disposable wedge sponges to apply makeup, you are using a lot more foundation than you would use otherwise. Based on these numbers, someone who uses disposable sponges will use 221.15g (7.801oz) of foundation per year, compared to the 75.45g (2.661oz) used by someone who applies with their fingers. Even if your foundation is drugstore prices, around $10 per ounce, you would save yourself $51.40 (plus the cost of the sponges) every year by switching. If you use fancier foundation, that difference may be much higher.
Looking at the data, I am confident that more statistically significant results would have come up if I had had more power (in other words, if I had been willing to apply and remove my foundation EVEN MORE FUCKING TIMES THAN I ALREADY DID). Unfortunately, my methods were super boring, so it is highly unlikely that I will be collecting any more data just to prove a point. It's really up to you to decide if these differences are enough to switch to a new method of application.
Additionally, it is worth stating explicitly that although the amount of foundation used may influence your makeup application decisions to some extent, there are lots of reasons to choose a more expensive method, including ease and speed of application, the finish, and sanitary issues.