One can use the same tablecloth for both meat and milk meals if the tablecloth is clean. 

Even though the [Shulchan Aruch Yoreh Deah 89:4][1] writes that you must change tablecloths between meat to dairy meals, the Pitchei Teshuvah elaborates that the reasoning no longer applies. 

This is what the Pitchei Teshuvah (89:8) writes:

>  עיין מ"ש רדב"ז ח"ב סי' תשכ"א שכתב שאין דברי הרב אלא כשהיו מניחין הבשר והגבינה על המפה ואז יש לחוש שמא ידבקו זה בזה אבל מנהגינו להביא כל מאכל על השלחן בקערות כו' הלכך אפי' לצאת ידי חומרת הרשב"א ז"ל בהפוך המפה לחוד ובהסרת הפתיתין סגי וכן נהגו ע"כ דבריו עיין שם:
> See what the Radbaz wrote (ח"ב סי' תשכ"א) that this only applied when they used to place meat and milk on the [actual] tablecloth, and so there was concern that they would stick to each other. But our custom is to bring any food [to the table] in bowls, etc. Thus, even through the Rashba's stringency it is enough to shake out the tablecloth [lit. turn over the tablecloth and remove the pieces], and this is how people act according to his words. 

In short, if you eat food on a plate, you can use the same tablecloth.