If the bed reminds the husband of his first wife and makes him think about her while with the 2nd wife, then he should not use that bed Nedarim 20b quoted in Shulchan Aruch Orach Chaim 240,2:
ולא תתורו אחרי לבבכם מכאן אמר רבי אל ישתה אדם בכוס זה ויתן עיניו בכוס אחר אמר רבינא לא נצרכא אלא דאפילו ב' נשיו
"And you shall not be swayed after your heart"(Bamidbar 15,39) Rebbi said one should not drink from this cup and think about another cup (euphamism for wife see Ran:שבשעה שנזקק עם אשתו לא יתן עיניו באשה אחרת אפי' היא אשתו) Ravina says even if they are both his wives.
If every bed looks the same to him (most people probably share this view), and its his bed (if it belongs to the first wife he should give it back), and his second wife does not care about using the same bed (if she did care she would make sure he gets a new bed nowadays cheaply purchased), then he can use his own bed for his second marriage (his first wife should not be coming to his house checking on him).